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binomial distribution (Definition)

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

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

The distribution function determined by the probability function $f(x)$ is called a Bernoulli distribution or binomial distribution.

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

\includegraphics[scale=0.75]{binom10p3}
\includegraphics[scale=0.75]{binom10p5}

The corresponding distribution function is

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

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.

We will use the moment generating function to calculate the mean and variance for the distribution. The mentioned function is

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

which simplifies to
\begin{displaymath}G(t)=(pe^t+q)^n.\end{displaymath}

Differentiating gives us
\begin{displaymath}G'(t)=n(pe^t+q)^{n-1}pe^t\end{displaymath}

and therefore the mean is
\begin{displaymath}\mu = E[X]=G'(0)=np.\end{displaymath}

Now for the variance we need the second derivative

\begin{displaymath}G”(t)=n(n-1)(pe^t+q)^{n-2} + n(pe^t+q)^{n-1}pe^t\end{displaymath}

so we get
\begin{displaymath}E[X^2]=G”(0)=n(n-1)p^2 + np\end{displaymath}

and finally the variance (recall $q=1-p$):
\begin{displaymath}\sigma^2 = E[X^2] - E[X]^2 = npq.\end{displaymath}

For large values of $n$, the binomial coefficients are hard to compute, however in this cases we can use either the Poisson distribution or the normal distribution to approximate the probabilities.



"binomial distribution" is owned by Mathprof. [ full author list (2) | owner history (2) ]
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See Also: binomial coefficient, binomial theorem, Bernoulli random variable

Other names:  Bernoulli distribution, binomial random variable, binomial probability function, Bernoulli random variable
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Cross-references: normal distribution, Poisson distribution, binomial coefficients, second derivative, function, variance, mean, moment generating function, binomial, calculate, probability function, distribution function, outcomes
There are 9 references to this entry.

This is version 14 of binomial distribution, born on 2002-09-13, modified 2007-05-18.
Object id is 3454, canonical name is BernoulliDistribution2.
Accessed 26722 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
Pending Errata and Addenda
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typo by Niels Diepeveen on 2007-03-09 22:59:47
I think I noticed a minor typo:
Under
 The corresponding distribution function is
it should read
 F(x) =
rather than
 F(n) =
unless I'm very much mistaken.
Otherwise: very useful, thank you.
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