Geometric distribution

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Geometric_distribution 

 

Probability mass function
Cumulative distribution function
Parameters	${\displaystyle 0<p\le 1}$ success probability (real)	${\displaystyle 0<p\le 1}$ success probability (real)
Support	k trials where ${\displaystyle k\in \{1,2,3,\dots \}}$	k failures where ${\displaystyle k\in \{0,1,2,3,\dots \}}$
PMF	${\displaystyle (1-p{)}^{k-1}p}$	${\displaystyle (1-p{)}^{k}p}$
CDF	${\displaystyle 1-(1-p{)}^{\lfloor x\rfloor }}$ for ${\displaystyle x\ge 1}$, ${\displaystyle 0}$ for ${\displaystyle x<1}$	${\displaystyle 1-(1-p{)}^{\lfloor x\rfloor +1}}$ for ${\displaystyle x\ge 0}$, ${\displaystyle 0}$ for ${\displaystyle x<0}$
Mean	${\displaystyle \frac{1}{p}}$	${\displaystyle \frac{1-p}{p}}$
Median	${\displaystyle \lceil \frac{-1}{{\log }_2(1-p)}\rceil }$ (not unique if ${\displaystyle -1/{\log }_2(1-p)}$ is an integer)	${\displaystyle \lceil \frac{-1}{{\log }_2(1-p)}\rceil -1}$ (not unique if ${\displaystyle -1/{\log }_2(1-p)}$ is an integer)
Mode	${\displaystyle 1}$	${\displaystyle 0}$
Variance	${\displaystyle \frac{1-p}{p^2}}$	${\displaystyle \frac{1-p}{p^2}}$
Skewness	${\displaystyle \frac{2-p}{\sqrt{1-p}}}$	${\displaystyle \frac{2-p}{\sqrt{1-p}}}$
Ex. kurtosis	${\displaystyle 6+\frac{p^2}{1-p}}$	${\displaystyle 6+\frac{p^2}{1-p}}$
Entropy	${\displaystyle \frac{-(1-p)\log (1-p)-p\log p}{p}}$	${\displaystyle \frac{-(1-p)\log (1-p)-p\log p}{p}}$
MGF	${\displaystyle \frac{p{e}^t}{1-(1-p)e^t},}$ for ${\displaystyle t<-\ln (1-p)}$	${\displaystyle \frac{p}{1-(1-p)e^t},}$ for ${\displaystyle t<-\ln (1-p)}$
CF	${\displaystyle \frac{p{e}^{it}}{1-(1-p)e^{it}}}$	${\displaystyle \frac{p}{1-(1-p)e^{it}}}$
PGF	${\displaystyle \frac{pz}{1-(1-p)z}}$	${\displaystyle \frac{p}{1-(1-p)z}}$

In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:

• The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set ${\displaystyle \{1,2,3,\dots \}}$;
• The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set ${\displaystyle \{0,1,2,\dots \}}$.

Which of these is called the geometric distribution is a matter of convention and convenience.

These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of the number X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.

The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial is the first success is

${\displaystyle Pr(X=k)=(1-p{)}^{k-1}p}$

for k = 1, 2, 3, 4, ....

The above form of the geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success:

${\displaystyle Pr(Y=k)=Pr(X=k+1)=(1-p{)}^{k}p}$

for k = 0, 1, 2, 3, ....

In either case, the sequence of probabilities is a geometric sequence.

For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set { 1, 2, 3, ... } and is a geometric distribution with p = 1/6.

The geometric distribution is denoted by Geo(p) where 0 < p ≤ 1. 

Definitions

Consider a sequence of trials, where each trial has only two possible outcomes (designated failure and success). The probability of success is assumed to be the same for each trial. In such a sequence of trials, the geometric distribution is useful to model the number of failures before the first success since the experiment can have an indefinite number of trials until success, unlike the binomial distribution which has a set number of trials. The distribution gives the probability that there are zero failures before the first success, one failure before the first success, two failures before the first success, and so on.

Assumptions: When is the geometric distribution an appropriate model?

The geometric distribution is an appropriate model if the following assumptions are true.

• The phenomenon being modelled is a sequence of independent trials.
• There are only two possible outcomes for each trial, often designated success or failure.
• The probability of success, p, is the same for every trial.

If these conditions are true, then the geometric random variable Y is the count of the number of failures before the first success. The possible number of failures before the first success is 0, 1, 2, 3, and so on. In the graphs above, this formulation is shown on the right.

An alternative formulation is that the geometric random variable X is the total number of trials up to and including the first success, and the number of failures is X − 1. In the graphs above, this formulation is shown on the left.

Probability outcomes examples

The general formula to calculate the probability of k failures before the first success, where the probability of success is p and the probability of failure is q = 1 − p, is

${\displaystyle Pr(Y=k)=q^{k}p.}$

for k = 0, 1, 2, 3, ...

E1) A doctor is seeking an antidepressant for a newly diagnosed patient. Suppose that, of the available anti-depressant drugs, the probability that any particular drug will be effective for a particular patient is p = 0.6. What is the probability that the first drug found to be effective for this patient is the first drug tried, the second drug tried, and so on? What is the expected number of drugs that will be tried to find one that is effective?

The probability that the first drug works. There are zero failures before the first success. Y = 0 failures. The probability Pr(zero failures before first success) is simply the probability that the first drug works.

${\displaystyle Pr(Y=0)=q^{0}p={0.4}^0\times 0.6=1\times 0.6=\mathrm{0.6.}}$

The probability that the first drug fails, but the second drug works. There is one failure before the first success. Y = 1 failure. The probability for this sequence of events is Pr(first drug fails) ${\displaystyle \times }$ p(second drug succeeds), which is given by

${\displaystyle Pr(Y=1)=q^{1}p={0.4}^1\times 0.6=0.4\times 0.6=\mathrm{0.24.}}$

The probability that the first drug fails, the second drug fails, but the third drug works. There are two failures before the first success. Y = 2 failures. The probability for this sequence of events is Pr(first drug fails) ${\displaystyle \times }$ p(second drug fails) ${\displaystyle \times }$ Pr(third drug is success)

${\displaystyle Pr(Y=2)=q^{2}p,={0.4}^2\times 0.6=\mathrm{0.096.}}$

E2) A newlywed couple plans to have children and will continue until the first girl. What is the probability that there are zero boys before the first girl, one boy before the first girl, two boys before the first girl, and so on?

The probability of having a girl (success) is p= 0.5 and the probability of having a boy (failure) is q = 1 − p = 0.5.

The probability of no boys before the first girl is

${\displaystyle Pr(Y=0)=q^{0}p={0.5}^0\times 0.5=1\times 0.5=\mathrm{0.5.}}$

The probability of one boy before the first girl is

${\displaystyle Pr(Y=1)=q^{1}p={0.5}^1\times 0.5=0.5\times 0.5=\mathrm{0.25.}}$

The probability of two boys before the first girl is

${\displaystyle Pr(Y=2)=q^{2}p={0.5}^2\times 0.5=\mathrm{0.125.}}$

and so on.

Properties

Moments and cumulants

The expected value for the number of independent trials to get the first success, and the variance of a geometrically distributed random variable X is:

${\displaystyle \mathrm{E}(X)=\frac{1}{p},\mathrm{var}(X)=\frac{1-p}{p^2}.}$

Similarly, the expected value and variance of the geometrically distributed random variable Y = X - 1 (See definition of distribution ${\displaystyle Pr(Y=k)}$) is:

${\displaystyle \mathrm{E}(Y)=\mathrm{E}(X-1)=\mathrm{E}(X)-1=\frac{1-p}{p},\mathrm{var}(Y)=\frac{1-p}{p^2}.}$

Proof

Expected value of X

Consider the expected value ${\displaystyle \mathrm{E}(X)}$ of X as above, i.e. the average number of trials until a success. On the first trial we either succeed with probability ${\displaystyle p}$, or we fail with probability ${\displaystyle 1-p}$. If we fail the remaining mean number of trials until a success is identical to the original mean. This follows from the fact that all trials are independent. From this we get the formula:

${\displaystyle \mathrm{E}(X)=p\cdot 1+(1-p)\cdot (1+\mathrm{E}(X)),}$

which if solved for ${\displaystyle \mathrm{E}(X)}$ gives:

${\displaystyle \mathrm{E}(X)=\frac{1}{p}.}$

Expected value of Y

That the expected value of Y as above is (1 − p)/p can be shown in the following way:

${\displaystyle \begin{array}{rl}\mathrm{E}(Y)& =\sum _{k=0}^{\mathrm{\infty }}(1-p{)}^{k}p\cdot k\\ & =p\sum _{k=0}^{\mathrm{\infty }}(1-p{)}^{k}k\\ & =p(1-p)\sum _{k=0}^{\mathrm{\infty }}(1-p{)}^{k-1}\cdot k\\ & =p(1-p)\left[\frac{d}{dp}\left(-\sum _{k=0}^{\mathrm{\infty }}(1-p{)}^k\right)\right]\\ & =p(1-p)\frac{d}{dp}\left(-\frac{1}{p}\right)=\frac{1-p}{p}.\end{array}}$

The interchange of summation and differentiation is justified by the fact that convergent power series converge uniformly on compact subsets of the set of points where they converge.

Let μ = (1 − p)/p be the expected value of Y. Then the cumulants ${\displaystyle {\kappa }_n}$ of the probability distribution of Y satisfy the recursion

${\displaystyle {\kappa }_{n+1}=\mu (\mu +1)\frac{d{\kappa }_n}{d\mu }.}$

Expected value examples

E3) A patient is waiting for a suitable matching kidney donor for a transplant. If the probability that a randomly selected donor is a suitable match is p = 0.1, what is the expected number of donors who will be tested before a matching donor is found?

With p = 0.1, the mean number of failures before the first success is E(Y) = (1 − p)/p =(1 − 0.1)/0.1 = 9.

For the alternative formulation, where X is the number of trials up to and including the first success, the expected value is E(X) = 1/p = 1/0.1 = 10.

For example 1 above, with p = 0.6, the mean number of failures before the first success is E(Y) = (1 − p)/p = (1 − 0.6)/0.6 = 0.67.

Higher-order moments

The moments for the number of failures before the first success are given by

${\displaystyle \begin{array}{rl}\mathrm{E}(Y^n)& =\sum _{k=0}^{\mathrm{\infty }}(1-p{)}^{k}p\cdot k^n\\ & =p{\mathrm{Li}}_{-n}(1-p)& (\text{for }n\ne 0)\end{array}}$

where ${\displaystyle {\mathrm{Li}}_{-n}(1-p)}$ is the polylogarithm function.

General properties

• The probability-generating functions of X and Y are, respectively,

${\displaystyle \begin{array}{rl}{G}_X(s)& =\frac{sp}{1-s(1-p)},\\ G_Y(s)& =\frac{p}{1-s(1-p)},|s|<(1-p{)}^{-1}.\end{array}}$

• Like its continuous analogue (the exponential distribution), the geometric distribution is memoryless. That is, the following holds for every m and n.

${\displaystyle Pr\{X>m+n|X>n\}=Pr\{X>m\}}$

The geometric distribution supported on {1, 2, 3, ... } is the only memoryless discrete distribution. Note that the geometric distribution supported on {0, 1, 2, ... } is not memoryless.

• Among all discrete probability distributions supported on {1, 2, 3, ... } with given expected value μ, the geometric distribution X with parameter p = 1/μ is the one with the largest entropy.
• The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y₁, ..., Y_n whose sum has the same distribution that Y has. These will not be geometrically distributed unless n = 1; they follow a negative binomial distribution.
• The decimal digits of the geometrically distributed random variable Y are a sequence of independent (and not identically distributed) random variables. For example, the hundreds digit D has this probability distribution:

${\displaystyle Pr(D=d)=\frac{q^{100d}}{1+q^{100}+q^{200}+\cdots +q^{900}},}$

where q = 1 − p, and similarly for the other digits, and, more generally, similarly for numeral systems with other bases than 10. When the base is 2, this shows that a geometrically distributed random variable can be written as a sum of independent random variables whose probability distributions are indecomposable.

• Golomb coding is the optimal prefix code for the geometric discrete distribution.
• The sum of two independent Geo(p) distributed random variables is not a geometric distribution. 

Related distributions

• The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y₁, ..., Y_r are independent geometrically distributed variables with parameter p, then the sum

${\displaystyle Z=\sum _{m=1}^r{Y}_m}$

follows a negative binomial distribution with parameters r and p.

• The geometric distribution is a special case of discrete compound Poisson distribution.
• If Y₁, ..., Y_r are independent geometrically distributed variables (with possibly different success parameters p_m), then their minimum

${\displaystyle W=\underset{m\in 1,\dots ,r}{min}{Y}_m}$

is also geometrically distributed, with parameter ${\displaystyle p=1-\prod _m(1-p_m).}$ 

• Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable X_k has a Poisson distribution with expected value r ^k/k. Then

${\displaystyle \sum _{k=1}^{\mathrm{\infty }}k{X}_k}$

has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value r/(1 − r).

• The exponential distribution is the continuous analogue of the geometric distribution. If X is an exponentially distributed random variable with parameter λ, then

${\displaystyle Y=\lfloor X\rfloor ,}$

where ${\displaystyle \lfloor \rfloor }$ is the floor (or greatest integer) function, is a geometrically distributed random variable with parameter p = 1 − e^−λ (thus λ = −ln(1 − p)) and taking values in the set {0, 1, 2, ...}. This can be used to generate geometrically distributed pseudorandom numbers by first generating exponentially distributed pseudorandom numbers from a uniform pseudorandom number generator: then ${\displaystyle \lfloor \ln (U)/\ln (1-p)\rfloor }$ is geometrically distributed with parameter ${\displaystyle p}$, if ${\displaystyle U}$ is uniformly distributed in [0,1].

• If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since

${\displaystyle \begin{array}{rl}Pr(X/n>a)=Pr(X>na)& =(1-p{)}^{na}={\left(1-\frac{1}{n}\right)}^{na}={\left[{\left(1-\frac{1}{n}\right)}^n\right]}^a\\ & \to [e^{-1}{]}^a=e^{-a}\text{ as }n\to \mathrm{\infty }.\end{array}}$

More generally, if p = λ/n, where λ is a parameter, then as n→ ∞ the distribution of X/n approaches an exponential distribution with rate λ:

${\displaystyle Pr(X>nx)=\underset{n\to \mathrm{\infty }}{lim}(1-\lambda /n{)}^{nx}=e^{-\lambda x}}$

therefore the distribution function of X/n converges to ${\displaystyle 1-e^{-\lambda x}}$, which is that of an exponential random variable.

Statistical inference

Parameter estimation

For both variants of the geometric distribution, the parameter p can be estimated by equating the expected value with the sample mean. This is the method of moments, which in this case happens to yield maximum likelihood estimates of p.

Specifically, for the first variant let k = k₁, ..., k_n be a sample where k_i ≥ 1 for i = 1, ..., n. Then p can be estimated as

${\displaystyle \hat{p}={\left(\frac{1}{n}\sum _{i=1}^n{k}_i\right)}^{-1}=\frac{n}{\sum _{i=1}^n{k}_i}.}$

In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta(α, β) prior, then the posterior distribution is

${\displaystyle p\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}\left(\alpha +n,\beta +\sum _{i=1}^n(k_i-1)\right).}$

The posterior mean E[p] approaches the maximum likelihood estimate ${\displaystyle \hat{p}}$ as α and β approach zero.

In the alternative case, let k₁, ..., k_n be a sample where k_i ≥ 0 for i = 1, ..., n. Then p can be estimated as

${\displaystyle \hat{p}={\left(1+\frac{1}{n}\sum _{i=1}^n{k}_i\right)}^{-1}=\frac{n}{\sum _{i=1}^n{k}_i+n}.}$

The posterior distribution of p given a Beta(α, β) prior is

${\displaystyle p\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}\left(\alpha +n,\beta +\sum _{i=1}^n{k}_i\right).}$

Again the posterior mean E[p] approaches the maximum likelihood estimate ${\displaystyle \hat{p}}$ as α and β approach zero.

For either estimate of ${\displaystyle \hat{p}}$ using Maximum Likelihood, the bias is equal to

${\displaystyle b\equiv \mathrm{E}[({\hat{p}}_{\mathrm{m}\mathrm{l}\mathrm{e}}-p)]=\frac{p(1-p)}{n}}$

which yields the bias-corrected maximum likelihood estimator

${\displaystyle {\hat{p}}_{\text{mle}}^{\ast }={\hat{p}}_{\text{mle}}-\hat{b}}$

Computational methods

Geometric distribution using R

The R function dgeom(k, prob) calculates the probability that there are k failures before the first success, where the argument "prob" is the probability of success on each trial.

For example,

dgeom(0,0.6) = 0.6

dgeom(1,0.6) = 0.24

R uses the convention that k is the number of failures, so that the number of trials up to and including the first success is k + 1.

The following R code creates a graph of the geometric distribution from Y = 0 to 10, with p = 0.6.

Y=0:10

plot(Y, dgeom(Y,0.6), type="h", ylim=c(0,1), main="Geometric distribution for p=0.6", ylab="Pr(Y=Y)", xlab="Y=Number of failures before first success")

Geometric distribution using Excel

The geometric distribution, for the number of failures before the first success, is a special case of the negative binomial distribution, for the number of failures before s successes.

The Excel function NEGBINOMDIST(number_f, number_s, probability_s) calculates the probability of k = number_f failures before s = number_s successes where p = probability_s is the probability of success on each trial. For the geometric distribution, let number_s = 1 success.

For example,

=NEGBINOMDIST(0, 1, 0.6) = 0.6

=NEGBINOMDIST(1, 1, 0.6) = 0.24

Like R, Excel uses the convention that k is the number of failures, so that the number of trials up to and including the first success is k + 1.

Beta particle

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Beta_particle

Alpha radiation consists of helium nuclei and is readily stopped by a sheet of paper. Beta radiation, consisting of electrons or positrons, is stopped by a thin aluminum plate, but gamma radiation requires shielding by dense material such as lead or concrete.

A beta particle, also called beta ray or beta radiation (symbol β), is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus during the process of beta decay. There are two forms of beta decay, β⁻ decay and β⁺ decay, which produce electrons and positrons respectively.

Beta particles with an energy of 0.5 MeV have a range of about one metre in the air; the distance is dependent on the particle energy.

Beta particles are a type of ionizing radiation and for radiation protection purposes are regarded as being more ionising than gamma rays, but less ionising than alpha particles. The higher the ionising effect, the greater the damage to living tissue, but also the lower the penetrating power of the radiation.

Beta decay modes

β⁻ decay (electron emission)

Main article: β⁻ decay

Beta decay. A beta particle (in this case a negative electron) is shown being emitted by a nucleus. An antineutrino (not shown) is always emitted along with an electron. Insert: in the decay of a free neutron, a proton, an electron (negative beta ray), and an electron antineutrino are produced.

An unstable atomic nucleus with an excess of neutrons may undergo β⁻ decay, where a neutron is converted into a proton, an electron, and an electron antineutrino (the antiparticle of the neutrino):

n
→
p
+
e⁻
+
ν
_e

This process is mediated by the weak interaction. The neutron turns into a proton through the emission of a virtual W⁻ boson. At the quark level, W⁻ emission turns a down quark into an up quark, turning a neutron (one up quark and two down quarks) into a proton (two up quarks and one down quark). The virtual W⁻ boson then decays into an electron and an antineutrino.

β− decay commonly occurs among the neutron-rich fission byproducts produced in nuclear reactors. Free neutrons also decay via this process. Both of these processes contribute to the copious quantities of beta rays and electron antineutrinos produced by fission-reactor fuel rods.

β⁺ decay (positron emission)

Main article: Positron emission

Unstable atomic nuclei with an excess of protons may undergo β⁺ decay, also called positron decay, where a proton is converted into a neutron, a positron, and an electron neutrino:

p
→
n
+
e⁺
+
ν
_e

Beta-plus decay can only happen inside nuclei when the absolute value of the binding energy of the daughter nucleus is greater than that of the parent nucleus, i.e., the daughter nucleus is a lower-energy state.

Beta decay schemes

Caesium-137 decay scheme, showing it initially undergoes beta decay. The 661 keV gamma peak associated with ¹³⁷Cs is actually emitted by the daughter radionuclide.

The accompanying decay scheme diagram shows the beta decay of caesium-137. ¹³⁷Cs is noted for a characteristic gamma peak at 661 KeV, but this is actually emitted by the daughter radionuclide ^137mBa. The diagram shows the type and energy of the emitted radiation, its relative abundance, and the daughter nuclides after decay.

Phosphorus-32 is a beta emitter widely used in medicine and has a short half-life of 14.29 days and decays into sulfur-32 by beta decay as shown in this nuclear equation:

32 15P

→

32 16S1+

+

e−

+

ν e

1.709 MeV of energy is released during the decay. The kinetic energy of the electron varies with an average of approximately 0.5 MeV and the remainder of the energy is carried by the nearly undetectable electron antineutrino. In comparison to other beta radiation-emitting nuclides, the electron is moderately energetic. It is blocked by around 1 m of air or 5 mm of acrylic glass.

Interaction with other matter

Blue Cherenkov radiation light being emitted from a TRIGA reactor pool is due to high-speed beta particles traveling faster than the speed of light (phase velocity) in water (which is 75% of the speed of light in vacuum).

Of the three common types of radiation given off by radioactive materials, alpha, beta and gamma, beta has the medium penetrating power and the medium ionising power. Although the beta particles given off by different radioactive materials vary in energy, most beta particles can be stopped by a few millimeters of aluminium. However, this does not mean that beta-emitting isotopes can be completely shielded by such thin shields: as they decelerate in matter, beta electrons emit secondary gamma rays, which are more penetrating than betas per se. Shielding composed of materials with lower atomic weight generates gammas with lower energy, making such shields somewhat more effective per unit mass than ones made of high-Z materials such as lead.

Being composed of charged particles, beta radiation is more strongly ionizing than gamma radiation. When passing through matter, a beta particle is decelerated by electromagnetic interactions and may give off bremsstrahlung x-rays.

In water, beta radiation from many nuclear fission products typically exceeds the speed of light in that material (which is 75% that of light in vacuum), and thus generates blue Cherenkov radiation when it passes through water. The intense beta radiation from the fuel rods of swimming pool reactors can thus be visualized through the transparent water that covers and shields the reactor (see illustration at right).

Detection and measurement

Beta radiation detected in an isopropanol cloud chamber (after insertion of an artificial source strontium-90)

The ionizing or excitation effects of beta particles on matter are the fundamental processes by which radiometric detection instruments detect and measure beta radiation. The ionization of gas is used in ion chambers and Geiger–Müller counters, and the excitation of scintillators is used in scintillation counters. The following table shows radiation quantities in SI and non-SI units:

Ionizing radiation related quantities

Quantity	Unit	Symbol	Derivation	Year	SI equivalent
Activity (A)	becquerel	Bq	s−1	1974	SI unit
	curie	Ci	3.7 × 1010 s−1	1953	3.7×1010 Bq
	rutherford	Rd	106 s−1	1946	1,000,000 Bq
Exposure (X)	coulomb per kilogram	C/kg	C⋅kg−1 of air	1974	SI unit
	röntgen	R	esu / 0.001293 g of air	1928	2.58 × 10−4 C/kg
Absorbed dose (D)	gray	Gy	J⋅kg−1	1974	SI unit
	erg per gram	erg/g	erg⋅g−1	1950	1.0 × 10−4 Gy
	rad	rad	100 erg⋅g−1	1953	0.010 Gy
Equivalent dose (H)	sievert	Sv	J⋅kg−1 × WR	1977	SI unit
	röntgen equivalent man	rem	100 erg⋅g−1 × WR	1971	0.010 Sv
Effective dose (E)	sievert	Sv	J⋅kg−1 × WR × WT	1977	SI unit
	röntgen equivalent man	rem	100 erg⋅g−1 × WR × WT	1971	0.010 Sv

• The gray (Gy), is the SI unit of absorbed dose, which is the amount of radiation energy deposited in the irradiated material. For beta radiation this is numerically equal to the equivalent dose measured by the sievert, which indicates the stochastic biological effect of low levels of radiation on human tissue. The radiation weighting conversion factor from absorbed dose to equivalent dose is 1 for beta, whereas alpha particles have a factor of 20, reflecting their greater ionising effect on tissue.
• The rad is the deprecated CGS unit for absorbed dose and the rem is the deprecated CGS unit of equivalent dose, used mainly in the USA.

Applications

Beta particles can be used to treat health conditions such as eye and bone cancer and are also used as tracers. Strontium-90 is the material most commonly used to produce beta particles.

Beta particles are also used in quality control to test the thickness of an item, such as paper, coming through a system of rollers. Some of the beta radiation is absorbed while passing through the product. If the product is made too thick or thin, a correspondingly different amount of radiation will be absorbed. A computer program monitoring the quality of the manufactured paper will then move the rollers to change the thickness of the final product.

An illumination device called a betalight contains tritium and a phosphor. As tritium decays, it emits beta particles; these strike the phosphor, causing the phosphor to give off photons, much like the cathode-ray tube in a television. The illumination requires no external power, and will continue as long as the tritium exists (and the phosphors do not themselves chemically change); the amount of light produced will drop to half its original value in 12.32 years, the half-life of tritium.

Beta-plus (or positron) decay of a radioactive tracer isotope is the source of the positrons used in positron emission tomography (PET scan).

History

Henri Becquerel, while experimenting with fluorescence, accidentally found out that uranium exposed a photographic plate, wrapped with black paper, with some unknown radiation that could not be turned off like X-rays.

Ernest Rutherford continued these experiments and discovered two different kinds of radiation:

• alpha particles that did not show up on the Becquerel plates because they were easily absorbed by the black wrapping paper
• beta particles which are 100 times more penetrating than alpha particles.

He published his results in 1899.

In 1900, Becquerel measured the mass-to-charge ratio (m/e) for beta particles by the method of J. J. Thomson used to study cathode rays and identify the electron. He found that e/m for a beta particle is the same as for Thomson's electron, and therefore suggested that the beta particle is in fact an electron.

Health

Beta particles are moderately penetrating in living tissue, and can cause spontaneous mutation in DNA.

Beta sources can be used in radiation therapy to kill cancer cells.

Aerobic fermentation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Aerobic_fermentation

Aerobic fermentation or aerobic glycolysis is a metabolic process by which cells metabolize sugars via fermentation in the presence of oxygen and occurs through the repression of normal respiratory metabolism. Preference of aerobic fermentation over aerobic respiration is referred to as the Crabtree effect in yeast, and is part of the Warburg effect in tumor cells. While aerobic fermentation does not produce adenosine triphosphate (ATP) in high yield, it allows proliferating cells to convert nutrients such as glucose and glutamine more efficiently into biomass by avoiding unnecessary catabolic oxidation of such nutrients into carbon dioxide, preserving carbon-carbon bonds and promoting anabolism.

Aerobic fermentation in yeast

Aerobic fermentation evolved independently in at least three yeast lineages (Saccharomyces, Dekkera, Schizosaccharomyces). It has also been observed in plant pollen, trypanosomatids, mutated E. coli, and tumor cells. Crabtree-positive yeasts will respire when grown with very low concentrations of glucose or when grown on most other carbohydrate sources. The Crabtree effect is a regulatory system whereby respiration is repressed by fermentation, except in low sugar conditions. When Saccharomyces cerevisiae is grown below the sugar threshold and undergoes a respiration metabolism, the fermentation pathway is still fully expressed, while the respiration pathway is only expressed relative to the sugar availability. This contrasts with the Pasteur effect, which is the inhibition of fermentation in the presence of oxygen and observed in most organisms.

The evolution of aerobic fermentation likely involved multiple successive molecular steps, which included the expansion of hexose transporter genes, copy number variation (CNV) and differential expression in metabolic genes, and regulatory reprogramming. Research is still needed to fully understand the genomic basis of this complex phenomenon. Many Crabtree-positive yeast species are used for their fermentation ability in industrial processes in the production of wine, beer, sake, bread, and bioethanol. Through domestication, these yeast species have evolved, often through artificial selection, to better fit their environment. Strains evolved through mechanisms that include interspecific hybridization, horizontal gene transfer (HGT), gene duplication, pseudogenization, and gene loss.

Origin of Crabtree effect in yeast

Approximately 100 million years ago (mya), within the yeast lineage there was a whole genome duplication (WGD). A majority of Crabtree-positive yeasts are post-WGD yeasts. It was believed that the WGD was a mechanism for the development of the Crabtree effect in these species due to the duplication of alcohol dehydrogenase (ADH) encoding genes and hexose transporters. However, recent evidence has shown that aerobic fermentation originated before the WGD and evolved as a multi-step process, potentially aided by the WGD. The origin of aerobic fermentation, or the first step, in Saccharomyces Crabtree-positive yeasts likely occurred in the interval between the ability to grow under anaerobic conditions, horizontal transfer of anaerobic DHODase (encoded by URA1 with bacteria), and the loss of respiratory chain Complex I. A more pronounced Crabtree effect, the second step, likely occurred near the time of the WGD event. Later evolutionary events that aided in the evolution of aerobic fermentation are better understood and outlined in the section discussing the genomic basis of the Crabtree effect.

Driving forces

It is believed that a major driving force in the origin of aerobic fermentation was its simultaneous origin with modern fruit (~125 mya). These fruits provided an abundance of simple sugar food source for microbial communities, including both yeast and bacteria. Bacteria, at that time, were able to produce biomass at a faster rate than the yeast. Producing a toxic compound, like ethanol, can slow the growth of bacteria, allowing the yeast to be more competitive. However, the yeast still had to use a portion of the sugar it consumes to produce ethanol. Crabtree-positive yeasts also have increased glycolytic flow, or increased uptake of glucose and conversion to pyruvate, which compensates for using a portion of the glucose to produce ethanol rather than biomass. Therefore, it is believed that the original driving force was to kill competitors. This is supported by research that determined the kinetic behavior of the ancestral ADH protein, which was found to be optimized to make ethanol, rather than consume it.

Further evolutionary events in the development of aerobic fermentation likely increased the efficiency of this lifestyle, including increased tolerance to ethanol and the repression of the respiratory pathway. In high sugar environments, S. cerevisiae outcompetes and dominants all other yeast species, except its closest relative Saccharomyces paradoxus. The ability of S. cerevisiae to dominate in high sugar environments evolved more recently than aerobic fermentation and is dependent on the type of high-sugar environment. Other yeasts' growth is dependent on the pH and nutrients of the high-sugar environment.

Genomic basis of the Crabtree effect

The genomic basis of the Crabtree effect is still being investigated, and its evolution likely involved multiple successive molecular steps that increased the efficiency of the lifestyle.

Expansion of hexose transporter genes

Hexose transporters (HXT) are a group of proteins that are largely responsible for the uptake of glucose in yeast. In S. cerevisiae, 20 HXT genes have been identified and 17 encode for glucose transporters (HXT1-HXT17), GAL2 encodes for a galactose transporter, and SNF3 and RGT2 encode for glucose sensors. The number of glucose sensor genes have remained mostly consistent through the budding yeast lineage, however glucose sensors are absent from Schizosaccharomyces pombe. Sch. pombe is a Crabtree-positive yeast, which developed aerobic fermentation independently from Saccharomyces lineage, and detects glucose via the cAMP-signaling pathway. The number of transporter genes vary significantly between yeast species and has continually increased during the evolution of the S. cerevisiae lineage. Most of the transporter genes have been generated by tandem duplication, rather than from the WGD. Sch. pombe also has a high number of transporter genes compared to its close relatives. Glucose uptake is believed to be a major rate-limiting step in glycolysis and replacing S. cerevisiae's HXT1-17 genes with a single chimera HXT gene results in decreased ethanol production or fully respiratory metabolism. Thus, having an efficient glucose uptake system appears to be essential to ability of aerobic fermentation. There is a significant positive correlation between the number of hexose transporter genes and the efficiency of ethanol production.

CNV in glycolysis genes

A scheme of transformation of glucose to alcohol by alcoholic fermentation.

After a WGD, one of the duplicated gene pair is often lost through fractionation; less than 10% of WGD gene pairs have remained in S. cerevisiae genome. A little over half of WGD gene pairs in the glycolysis reaction pathway were retained in post-WGD species, significantly higher than the overall retention rate. This has been associated with an increased ability to metabolize glucose into pyruvate, or higher rate of glycolysis. After glycolysis, pyruvate can either be further broken down by pyruvate decarboxylase (Pdc) or pyruvate dehydrogenase (Pdh). The kinetics of the enzymes are such that when pyruvate concentrations are high, due to a high rate of glycolysis, there is increased flux through Pdc and thus the fermentation pathway. The WGD is believed to have played a beneficial role in the evolution of the Crabtree effect in post-WGD species partially due to this increase in copy number of glycolysis genes.

CNV in fermentation genes

The fermentation reaction only involves two steps. Pyruvate is converted to acetaldehyde by Pdc and then acetaldehyde is converted to ethanol by alcohol dehydrogenase (Adh). There is no significant increase in the number of Pdc genes in Crabtree-positive compared to Crabtree-negative species and no correlation between number of Pdc genes and efficiency of fermentation. There are five Adh genes in S. cerevisiae. Adh1 is the major enzyme responsible for catalyzing the fermentation step from acetaldehyde to ethanol. Adh2 catalyzes the reverse reaction, consuming ethanol and converting it to acetaldehyde. The ancestral, or original, Adh had a similar function as Adh1 and after a duplication in this gene, Adh2 evolved a lower K_M for ethanol. Adh2 is believed to have increased yeast species' tolerance for ethanol and allowed Crabtree-positive species to consume the ethanol they produced after depleting sugars. However, Adh2 and consumption of ethanol is not essential for aerobic fermentation. Sch. pombe and other Crabtree positive species do not have the ADH2 gene and consumes ethanol very poorly.

Differential expression

In Crabtree-negative species, respiration related genes are highly expressed in the presence of oxygen. However, when S. cerevisiae is grown on glucose in aerobic conditions, respiration-related gene expression is repressed. Mitochondrial ribosomal proteins expression is only induced under environmental stress conditions, specifically low glucose availability. Genes involving mitochondrial energy generation and phosphorylation oxidation, which are involved in respiration, have the largest expression difference between aerobic fermentative yeast species and respiratory species. In a comparative analysis between Sch. pombe and S. cerevisiae, both of which evolved aerobic fermentation independently, the expression pattern of these two fermentative yeasts were more similar to each other than a respiratory yeast, C. albicans. However, S. cerevisiae is evolutionarily closer to C. albicans. Regulatory rewiring was likely important in the evolution of aerobic fermentation in both lineages.

Domestication and aerobic fermentation

A close up picture of ripening wine grapes. The light white "dusting" is a film that also contains wild yeasts.

Aerobic fermentation is essential for multiple industries, resulting in human domestication of several yeast strains. Beer and other alcoholic beverages, throughout human history, have played a significant role in society through drinking rituals, providing nutrition, medicine, and uncontaminated water. During the domestication process, organisms shift from natural environments that are more variable and complex to simple and stable environments with a constant substrate. This often favors specialization adaptations in domesticated microbes, associated with relaxed selection for non-useful genes in alternative metabolic strategies or pathogenicity. Domestication might be partially responsible for the traits that promote aerobic fermentation in industrial species. Introgression and HGT is common in Saccharomyces domesticated strains. Many commercial wine strains have significant portions of their DNA derived from HGT of non-Saccharomyces species. HGT and introgression are less common in nature than is seen during domestication pressures. For example, the important industrial yeast strain Saccharomyces pastorianus is an interspecies hybrid of S. cerevisiae and the cold tolerant S. eubayanus. This hybrid is commonly used in lager-brewing, which requires slow, low temperature fermentation.

Aerobic fermentation in acetic acid bacteria

See also: Acetic acid § Oxidative fermentation

Acetic acid bacteria (AAB) incompletely oxidize sugars and alcohols, usually glucose and ethanol, to acetic acid, in a process called AAB oxidative fermentation (AOF). After glycolysis, the produced pyruvate is broken down to acetaldehyde by pyruvate decarboxylase, which in turn is oxidized to acetic acid by acetaldehyde dehydrogenase. Ethanol is first oxidized to acetaldehyde by alcohol dehydrogenase, which is then converted to acetic acid. Both of these processes either generate NAD(P)H, or shuttle electrons into the electron transport chain via ubiquinol. This process is exploited in the use of acetic acid bacteria to produce vinegar.

Tumor cells

One of the hallmarks of cancer is altered metabolism or deregulating cellular energetics. Cancers cells often have reprogrammed their glucose metabolism to perform lactic acid fermentation, in the presence of oxygen, rather than send the pyruvate made through glycolysis to the mitochondria. This is referred to as the Warburg effect and is associated with high consumption of glucose and a high rate of glycolysis. ATP production in these cancer cells is often only through the process of glycolysis and pyruvate is broken down by the fermentation process in the cell's cytoplasm.

This phenomenon is often seen as counterintuitive, since cancer cells have higher energy demands due to the continued proliferation and respiration produces significantly more ATP than glycolysis alone (fermentation produces no additional ATP). Typically, there is an up-regulation in glucose transporters and enzymes in the glycolysis pathway (also seen in yeast). There are many parallel aspects of aerobic fermentation in tumor cells that are also seen in Crabtree-positive yeasts. Further research into the evolution of aerobic fermentation in yeast such as S. cerevisiae can be a useful model for understanding aerobic fermentation in tumor cells. This has a potential for better understanding cancer and cancer treatments.

Aerobic fermentation in other non-yeast species

Plants

Alcoholic fermentation is often used by plants in anaerobic conditions to produce ATP and regenerate NAD⁺ to allow for glycolysis to continue. For most plant tissues, fermentation only occurs in anaerobic conditions, but there are a few exceptions. In the pollen of maize (Zea mays) and tobacco (Nicotiana tabacum & Nicotiana plumbaginifolia), the fermentation enzyme ADH is abundant, regardless of the oxygen level. In tobacco pollen, PDC is also highly expressed in this tissue and transcript levels are not influenced by oxygen concentration. Tobacco pollen, similar to Crabtree-positive yeast, perform high levels of fermentation dependent on the sugar supply, and not oxygen availability. In these tissues, respiration and alcoholic fermentation occur simultaneously with high sugar availability. Fermentation produces the toxic acetaldehyde and ethanol, that can build up in large quantities during pollen development. It has been hypothesized that acetaldehyde is a pollen factor that causes cytoplasmic male sterility. Cytoplasmic male sterility is a trait observed in maize, tobacco and other plants in which there is an inability to produce viable pollen. It is believed that this trait might be due to the expression of the fermentation genes, ADH and PDC, a lot earlier on in pollen development than normal and the accumulation of toxic aldehyde.

Trypanosomatids

When grown in glucose-rich media, trypanosomatid parasites degrade glucose via aerobic fermentation. In this group, this phenomenon is not a pre-adaptation to/or remnant of anaerobic life, shown through their inability to survive in anaerobic conditions. It is believed that this phenomenon developed due to the capacity for a high glycolytic flux and the high glucose concentrations of their natural environment. The mechanism for repression of respiration in these conditions is not yet known.

E. coli mutants

A couple of Escherichia coli mutant strains have been bioengineered to ferment glucose under aerobic conditions. One group developed the ECOM3 (E. coli cytochrome oxidase mutant) strain by removing three terminal cytochrome oxidases (cydAB, cyoABCD, and cbdAB) to reduce oxygen uptake. After 60 days of adaptive evolution on glucose media, the strain displayed a mixed phenotype. In aerobic conditions, some populations' fermentation solely produced lactate, while others performed mixed-acid fermentation.