Binomial

The binomial distribution functions are functions of a nonnegative integer x,

$\displaystyle \func$BinomialDist(x;n, p) = $\displaystyle \sum_{{k=0}}^{{x}}$$\displaystyle \binom{n}{k}$pkqn-k

with Bernoulli trial parameter (or sample size) a positive integer n, Bernoulli probability parameter a real number p with 0 < p < 1 and q = 1 - p. (To enter binomial coefficients, $\binom{n}{k}$ = ${\frac{{n!}}{{k!\left( n-k\right) !}}}$, click the binomial fraction itbpF0.2992in0.2992in0.0692inbinomial.wmfand choose None for line.) The corresponding binomial probability density function is

$\displaystyle \func$BinomialDen(x;n, p) = $\displaystyle \binom{n}{x}$pxqn-x

for the same conditions on x, n, and p. The mean for this distribution is np, and the variance is npq.


Binomial distribution tables found in statistics books give selected values of either the binomial probability density function $\func$BinomialDen(x;n, p) or the cumulative distribution function $\func$BinomialDist(x;n, p).

The binomial density $\func$BinomialDen(x;n, p) gives the probability of x successes in n independent Bernoulli trials, when the probability of success at each trial is p. It is by far the most common discrete distribution, since people deal with many experiments in which a dichotomous classification of the result is of primary interest. The name binomial distribution comes from the fact that the coefficients $\binom{n}{k}$ = ${\frac{{n!}}{{k!\left( n-k\right) !}}}$ are commonly called binomial coefficients.



\begin{example} Compute the probability that, in $100$\ tosses of a coin with $... ...ion. \par $\Pr (X\leq 54)=\func{BinomialDist}(54;100,.55)=.45846$ \end{example}

The binomial distribution function with parameters n and p can be approximated by the normal distribution with mean np and variance np$\left(\vphantom{ 1-p}\right.$1 - p$\left.\vphantom{ 1-p}\right)$; that is,

$\displaystyle \func$BinomialDist(x;np, np(1 - p)) $\displaystyle \approx$ $\displaystyle \func$NormalDist(x;n, p)

Such approximations are reasonably good if np and n$\left(\vphantom{ 1-p}\right.$1 - p$\left.\vphantom{ 1-p}\right)$ are greater than 5. For example, to find an approximate solution to the preceding problem using a normal distribution, use


Pr(X ≤ 54) = $\displaystyle \func$BinomialDist(54;100,.55) = .45846  
  $\displaystyle \approx$ $\displaystyle \func$NormalDist(54;55.0, 24.75) = .48389  

The following plot shows the graph of $\func$NormalDist(x;55.0, 24.75) and a polygonal plot of $\func$BinomialDist(x;100,.55) for 0≤x≤100.

dtbpFU3in2.0003in0ptNormal and Binomial distributions