binomial

binomial distribution probabilities

Calling Sequence

pr=binomial(p,n)

Arguments

:pr row vector with n+1 components : :p real number in [0,1] : :n an integer >= 1 :

Description

pr=binomial(p,n) returns the binomial probability vector, i.e. pr(k+1) is the probability of k success in n independent Bernouilli trials with probability of success p. In other words : pr(k+1) = probability(X=k) , with X a random variable following the B(n,p) distribution, and numerically :

Examples

// first example
n=10;p=0.3; `clf`_(); `plot2d3`_(0:n,binomial(p,n));

// second example
n=50;p=0.4;
mea=n*p; sigma=`sqrt`_(n*p*(1-p));
x=( (0:n)-mea )/sigma;
`clf`_()
`plot2d`_(x, sigma*binomial(p,n));
`deff`_('y=Gauss(x)','y=1/sqrt(2*%pi)*exp(-(x.^2)/2)')
`plot2d`_(x, Gauss(x), style=2);

// by binomial formula (Caution if big n)
function pr=binomial2(p, n)
x=`poly`_(0,'x');pr=`coeff`_((1-p+x)^n).*`horner`_(x^(0:n),p);
endfunction
p=1/3;n=5;
binomial(p,n)-binomial2(p,n)

// by Gamma function: gamma(n+1)=n! (Caution if big n)
p=1/3;n=5;
Cnks=`gamma`_(n+1)./(`gamma`_(1:n+1).*`gamma`_(n+1:-1:1));
x=`poly`_(0,'x');
pr=Cnks.*`horner`_(x.^(0:n).*(1-x)^(n:-1:0),p);
pr-binomial(p,n)

See Also

• cdfbin cumulative distribution function Binomial distribution
• grand Random numbers