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


.. _grand: grand.html
.. _cdfbin: cdfbin.html