توزيع هندسي

Geometric

Probability mass function
Cumulative distribution function
المتغيرات	${\displaystyle{\displaystyle 0<p\leq 1}}$ success probability (real)	${\displaystyle{\displaystyle 0<p\leq 1}}$ success probability (real)
Support	${\displaystyle{\displaystyle k\in\{1,2,3,\dots\}\!}}$	${\displaystyle{\displaystyle k\in\{0,1,2,3,\dots\}\!}}$
Probability mass function (pmf)	${\displaystyle{\displaystyle(1-p)^{k-1}\,p\!}}$	${\displaystyle{\displaystyle(1-p)^{k}\,p\!}}$
Cumulative distribution function (cdf)	${\displaystyle{\displaystyle 1-(1-p)^{k}\!}}$	${\displaystyle{\displaystyle 1-(1-p)^{k+1}\!}}$
Mean	${\displaystyle{\displaystyle{\frac{1}{p}}\!}}$	${\displaystyle{\displaystyle{\frac{1-p}{p}}\!}}$
Median	${\displaystyle{\displaystyle\left\lceil{\frac{-1}{\log_{2}(1-p)}}\right\rceil\!}}$ (not unique if ${\displaystyle{\displaystyle-1/\log_{2}(1-p)}}$ is an integer)	${\displaystyle{\displaystyle\left\lceil{\frac{-1}{\log_{2}(1-p)}}\right\rceil\!-1}}$ (not unique if ${\displaystyle{\displaystyle-1/\log_{2}(1-p)}}$ is an integer)
Mode	${\displaystyle{\displaystyle 1}}$	${\displaystyle{\displaystyle 0}}$
Variance	${\displaystyle{\displaystyle{\frac{1-p}{p^{2}}}\!}}$	${\displaystyle{\displaystyle{\frac{1-p}{p^{2}}}\!}}$
Skewness	${\displaystyle{\displaystyle{\frac{2-p}{\sqrt{1-p}}}\!}}$	${\displaystyle{\displaystyle{\frac{2-p}{\sqrt{1-p}}}\!}}$
Excess kurtosis	${\displaystyle{\displaystyle 6+{\frac{p^{2}}{1-p}}\!}}$	${\displaystyle{\displaystyle 6+{\frac{p^{2}}{1-p}}\!}}$
Entropy	${\displaystyle{\displaystyle{\tfrac{-(1-p)\log_{2}(1-p)-p\log_{2}p}{p}}\!}}$	${\displaystyle{\displaystyle{\tfrac{-(1-p)\log_{2}(1-p)-p\log_{2}p}{p}}\!}}$
Moment-generating function (mgf)	${\displaystyle{\displaystyle{\frac{pe^{t}}{1-(1-p)e^{t}}}\!}}$, for ${\displaystyle{\displaystyle t<-\ln(1-p)\!}}$	${\displaystyle{\displaystyle{\frac{p}{1-(1-p)e^{t}}}\!}}$
Characteristic function	${\displaystyle{\displaystyle{\frac{pe^{it}}{1-(1-p)\,e^{it}}}\!}}$	${\displaystyle{\displaystyle{\frac{p}{1-(1-p)\,e^{it}}}\!}}$

التوزيع الهندسي Geometric distribution وهو جزء من التوزيع الاحتمالي الغير متعلق بمحاولات برنولي Bernoulli trials. ويستخدم التوزيع الهندسي كم عدد المحاولات التي نحتاجها للحصول على النتيجة المطلوبة ${\displaystyle{\displaystyle P(W)=p*(1-p)^{k-1}}}$

مثال ليكن لدينا نرد متجانس (1,2,3,4,5,6,) كم عدد المحاولات (n) التي نحتاجها للحصول على الرقم 6 الحل :

الاحتمال الصحيح P = 1/6

الاحتمالات الخاطئة q=1-P = 5/6

${\displaystyle{\displaystyle P(W)=p(1-p)^{n-1}=pq^{n-1}(n=1,2)}}$

${\displaystyle{\displaystyle p(w)=(1/5)*(5/6)^{(}n-1)}}$

 العزوم والتراكمات

The expected value of a geometrically distributed random variable X is 1/p and the variance is (1 − p)/p²:

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

Similarly, the expected value of the geometrically distributed random variable Y is (1 − p)/p, and its variance is (1 − p)/p²:

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

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

${\displaystyle{\displaystyle\kappa_{n+1}=\mu(\mu+1){\frac{d\kappa_{n}}{d\mu}}.}}$

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

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

(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.)

 خواص أخرى

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

${\displaystyle{\displaystyle{\begin{aligned} \displaystyle G_{X}(s)&\displaystyle={\frac{s\,p}{1-s\,(1-p)}},\\ \displaystyle G_{Y}(s)&\displaystyle={\frac{p}{1-s\,(1-p)}},\quad|s|<(1-p)^{-1}.\end{aligned}}}}$

• Golomb coding is the optimal prefix code for the geometric discrete distribution.

 توزيعات ذات صلة

• 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

${\displaystyle{\displaystyle Z=\sum_{m=1}^{r}Y_{m}}}$

follows a negative binomial distribution with parameters r and '1-'p.

• If Y₁, ..., Y_r are independent geometrically distributed variables (with possibly different success parameters p_m), then their minimum

${\displaystyle{\displaystyle W=\min_{m\in 1,\dots,r}Y_{m}\,}}$

is also geometrically distributed, with parameter ${\displaystyle{\displaystyle p=1-\prod_{m}(1-p_{m}).}}$

• 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

${\displaystyle{\displaystyle\sum_{k=1}^{\infty}k\,X_{k}}}$

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

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

where ${\displaystyle{\displaystyle\lfloor\quad\rfloor}}$ 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 ${\displaystyle{\displaystyle\lfloor\ln(U)/\ln(1-p)\rfloor}}$ is geometrically distributed with parameter ${\displaystyle{\displaystyle p}}$, if ${\displaystyle{\displaystyle U}}$ is uniformly distributed in [0,1].

 انظر أيضاً

• توزيع هندسي زائدي
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 وصلات خارجية

• Geometric distribution على بلانيت ماث
• Geometric distribution on MathWorld.
• Online geometric distribution calculator