توزيع ثنائي الحدين

Binomial distribution
	Probability mass function;
	Cumulative distribution function;
المتغيرات	– number of trials; – success probability for each trial
Support	– number of successes
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median	or
Mode	or
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

Binomial distribution for

{\displaystyle 70/256

توزيع احتمالي ثنائي هوتوزيع لتجربة عشوائية لها ناتجان فقط أحدهما نجاح التجربة والآخر فشلها ويكون الشرط الأساسي حتى احتمال النجاح لا يتأثر بتكرار التجربة ، أمثلة : رمي بترة نقود ، الإحصاءات أوالأسئلة التي تعتمد الإجابة لا أونعم.

بتعبير آخر التوزيع الاحتمالي ثنائي الحد هوتكرار لتجربة برنولي (انظر توزيع برنولي).

خصائص التوزيع الثنائي

يتميز التوزيع الثنائى بعدة خصائص هي:

تتكون التجربة من أكثر من محاولة. إذا تكونت التجربة من محاولة واحدة ،فإننا في تجربة توزيع برنولي.
استقلال المحاولات عن بعضها البعض أي ثبات احتمال النجاح p ومن ثم احتمال الفشل q.
هذه المحاولات جميعا متماثلة ومستقلة.
احتمال النجاح ثابت في جميع محاولة.

نطقب:بعض التوزيعات الاحتمالية الشائعة بمتغير واحد

{\displaystyle {\begin{aligned F(k;n,p)&=\Pr(X\leq k)\\&=I_{1-p (n-k,k+1)\\&=(n-k){n \choose k \int _{0 ^{1-p t^{n-k-1 (1-t)^{k \,dt.\end{aligned

Some closed-form bounds for the cumulative distribution function are given below.

Example

Suppose a biased coin comes up heads with probability 0.3 when tossed. What is the probability of achieving 0, 1,...,ستة heads after six tosses?

{\displaystyle \Pr(0{\text{ heads )=f(0)=\Pr(X=0)={6 \choose 0 0.3^{0 (1-0.3)^{6-0 =0.117649

{\displaystyle \Pr(1{\text{ heads )=f(1)=\Pr(X=1)={6 \choose 1 0.3^{1 (1-0.3)^{6-1 =0.302526

{\displaystyle \Pr(2{\text{ heads )=f(2)=\Pr(X=2)={6 \choose 2 0.3^{2 (1-0.3)^{6-2 =0.324135

{\displaystyle \Pr(3{\text{ heads )=f(3)=\Pr(X=3)={6 \choose 3 0.3^{3 (1-0.3)^{6-3 =0.18522

{\displaystyle \Pr(4{\text{ heads )=f(4)=\Pr(X=4)={6 \choose 4 0.3^{4 (1-0.3)^{6-4 =0.059535

{\displaystyle \Pr(5{\text{ heads )=f(5)=\Pr(X=5)={6 \choose 5 0.3^{5 (1-0.3)^{6-5 =0.010206

{\displaystyle \Pr(6{\text{ heads )=f(6)=\Pr(X=6)={6 \choose 6 0.3^{6 (1-0.3)^{6-6 =0.000729

Mean

If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of X is:

{\displaystyle \operatorname {E [X]=np.

For example, if n = 100, and p = 1/4, then the average number of successful results will be 25.

Proof: We calculate the mean, μ, directly calculated from its definition

{\displaystyle \mu =\sum _{i=0 ^{n x_{i p_{i ,

and the binomial theorem:

{\displaystyle {\begin{aligned \mu &=\sum _{k=0 ^{n k{\binom {n {k p^{k (1-p)^{n-k \\&=np\sum _{k=0 ^{n k{\frac {(n-1)! {(n-k)!k! p^{k-1 (1-p)^{(n-1)-(k-1) \\&=np\sum _{k=1 ^{n {\frac {(n-1)! {((n-1)-(k-1))!(k-1)! p^{k-1 (1-p)^{(n-1)-(k-1) \\&=np\sum _{k=1 ^{n {\binom {n-1 {k-1 p^{k-1 (1-p)^{(n-1)-(k-1) \\&=np\sum _{\ell =0 ^{n-1 {\binom {n-1 {\ell p^{\ell (1-p)^{(n-1)-\ell &&{\text{with \ell :=k-1\\&=np\sum _{\ell =0 ^{m {\binom {m {\ell p^{\ell (1-p)^{m-\ell &&{\text{with m:=n-1\\&=np(p+(1-p))^{m \\&=np\end{aligned

History

This distribution was derived by James Bernoulli. He considered the case where p = r/(r + s) where p is the probability of success and r and s are positive integers. Blaise Pascal had earlier considered the case where p = 1/2.

الهامش

^ Hamilton Institute. "The Binomial Distribution" October 20, 2010.
^ See Proof Wiki
^ Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A multifractal model of asset returns. 3.2 The Binomial Measure is the Simplest Example of a Multifractal

مراجع

https://www.jmasi.com/ehsa/prob/binomail.htm
https://faculty.ksu.edu.sa/mhmmurad/DocLib4/%D8%A7%D9%84%D9%88%D8%AD%D8%AF%D8%A9%20%D8%A7%D9%84%D8%AE%D8%A7%D9%85%D8%B3%D8%A9%20%D8%A7%D9%84%D8%AC%D8%B2%D8%A1%20%D8%A7%D9%84%D8%A7%D8%AB%D8%A7%D9%86%D9%8A.pdf

[1] Hamilton Institute. "The Binomial Distribution" October 20, 2010.

[2] See Proof Wiki

[3] Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A multifractal model of asset returns. 3.2 The Binomial Measure is the Simplest Example of a Multifractal

Probability mass function
Cumulative distribution function
المتغيرات	${\displaystyle n\in \{0,1,2,\ldots \$ – number of trials ${\displaystyle p\in [0,1]$ – success probability for each trial
Support	${\displaystyle k\in \{0,1,\ldots ,n\$ – number of successes
Probability mass function (pmf)	${\displaystyle {\binom {n {k p^{k (1-p)^{n-k$
Cumulative distribution function (cdf)	${\displaystyle I_{1-p (n-k,1+k)$
Mean	${\displaystyle np$
Median	${\displaystyle \lfloor np\rfloor$ or ${\displaystyle \lceil np\rceil$
Mode	${\displaystyle \lfloor (n+1)p\rfloor$ or ${\displaystyle \lceil (n+1)p\rceil -1$
Variance	${\displaystyle np(1-p)$
Skewness	${\displaystyle {\frac {1-2p {\sqrt {np(1-p)$
Excess kurtosis	${\displaystyle {\frac {1-6p(1-p) {np(1-p)$
Entropy	${\displaystyle {\frac {1 {2 \log _{2 \left(2\pi enp(1-p)\right)+O\left({\frac {1 {n \right)$
Moment-generating function (mgf)	${\displaystyle (1-p+pe^{t )^{n$
Characteristic function	${\displaystyle (1-p+pe^{it )^{n$