binomial distribution

Consider an experiment with two possible outcomes (success and failure), which happen randomly. Let $p$ be the probability of success. If the experiment is repeated $n$ times, the probability of having exactly $x$ successes is

 $f(x)=\left({n\atop x}\right)p^{x}(1-p)^{(n-x)}.$

Here are some plots for $f(x)$ with $n=20$ and $p=0.3$, $p=0.5$.  The corresponding distribution function is

 $F(x)=\sum_{k\leq x}\left({n\atop k}\right)p^{k}q^{n-k}$

where $q=1-p$. Notice that if we calculate $F(n)$ we get the binomial expansion for $(p+q)^{n}$, and this is the reason for the distribution  being called binomial.

 $G(t)=\sum_{x=0}^{n}e^{tx}\left({n\atop x}\right)p^{x}q^{n-x}$

which simplifies to

 $G(t)=(pe^{t}+q)^{n}.$

Differentiating gives us

 $G^{\prime}(t)=n(pe^{t}+q)^{n-1}pe^{t}$

and therefore the mean is

 $\mu=E[X]=G^{\prime}(0)=np.$

Now for the variance we need the second derivative

 $G^{\prime\prime}(t)=n(n-1)(pe^{t}+q)^{n-2}+n(pe^{t}+q)^{n-1}pe^{t}$

so we get

 $E[X^{2}]=G^{\prime\prime}(0)=n(n-1)p^{2}+np$

and finally the variance (recall $q=1-p$):

 $\sigma^{2}=E[X^{2}]-E[X]^{2}=npq.$
 Title binomial distribution Canonical name BinomialDistribution Date of creation 2013-03-22 13:03:01 Last modified on 2013-03-22 13:03:01 Owner Mathprof (13753) Last modified by Mathprof (13753) Numerical id 17 Author Mathprof (13753) Entry type Definition Classification msc 60E05 Synonym Bernoulli distribution Synonym binomial random variable  Synonym binomial probability function Synonym Bernoulli random variable Related topic BinomialCoefficient Related topic BinomialTheorem Related topic BernoulliRandomVariable