توزيع هندسي

Geometric
	Probability mass function;
	Cumulative distribution function;

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

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

الاحتمال السليم P = 1/6

الاحتمالات الخاطئة q=1-P = 5/6

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

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

العزوم والتراكمات

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

{\displaystyle \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 \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 ${\displaystyle \kappa _{n$ of the probability distribution of Y satisfy the recursion

{\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 {\begin{aligned \mathrm {E (Y)&{ =\sum _{k=0 ^{\infty (1-p)^{k p\cdot k\\&{ =p\sum _{k=0 ^{\infty (1-p)^{k k\\&{ =p\left[{\frac {d {dp \left(-\sum _{k=0 ^{\infty (1-p)^{k \right)\right](1-p)\\&{ =-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 {\begin{aligned G_{X (s)&={\frac {s\,p {1-s\,(1-p) ,\\[10pt]G_{Y (s)&={\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 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 W=\min _{m\in 1,\dots ,r 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 ^{\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 \ln(U)/\ln(1-p)\rfloor

انظر أيضاً

توزيع هندسي زائدي
مشكلة جامع الكوبونات

الهامش

^ http://www.wolframalpha.com/input/?i=inverse+p+%3D+1+-+e^-l

وصلات خارجية

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

نطقب:Common univariate probability distributions

[1] ttp://www.wolframalpha.com/input/?i=inverse+p+%3D+1+-+e^-l

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