   7.4.1  The binomial distribution

The probability density function for the binomial distribution : binomial

If you perform an experiment n times, where the probability of success each time is p, then the probability of exactly k successes is

binomial(n,k,p) =
 n k

pk (1−p)nk

This determines the binomial distribution, and so this is called the binomial command. If you enter

binomial(10,2,0.4)

you will get

0.120932352

If no third argument p is given, then binomial will just compute (kn), which recall is called the binomial coefficient and is also computed by comb. If you enter

binomial(10,2)

or

comb(10,2)

then you will get

45

The cumulative distribution function for the binomial distribution: binomial_cdf

Recall that the cumulative distribution function (cdf) for a distribution is cdf(x) = Prob(Xx). For the binomial distribution, this is given by the binomial_cdf command; binomial_cdf(n,p,x), which in this case will equal binomial(n,0,p) + …+ binomial(n,floor(x,p). If you enter

binomial_cdf(4,0.5,2)

you will get

0.6875

You can give binomial_cdf an additional argument; binomial_cdf(n,p,x,y) = Prob(xXy), which in this case would be binomial(n,ceil(x),p) + + binomial(n,floor(y),p). If you enter

binomial_cdf(2,0.3,1,2)

you will get

0.51

The inverse distribution function for the binomial distribution: binomial_icdf

Given a value h, the inverse distribution function gives the value of x so that Prob(Xx) = h; or for discrete distributions, the smallest x so that Prob(Xx) ≥ h. For the binomial distribution with n and p, the binomial_icdf gives the inverse distribution function. If you enter

binomial_icdf(4,0.5,0.9)

you will get

3

Note that binomial_cdf(4,0.5,3) is 0.9375, bigger than 0.9, while binomial_cdf(4,0.5,2) is 0.6875, smaller than 0.9.   