A random variable (X) has range (R=left{x_{1}, x_{2}, cdots, x_{m} ight}) and probability distribution [left{p_{k}=Pleft(X=x_{k} ight) ;

A random variable \(X\) has range \(R=\left\{x_{1}, x_{2}, \cdots, x_{m}\right\}\) and probability distribution

\[\left\{p_{k}=P\left(X=x_{k}\right) ; k=1,2, \ldots, m\right\}, \quad \Sigma_{k=1}^{m} p_{k}=1\]

A random experiment with outcome \(X\) is repeated \(n\) times. The outcome of the \(k t h\) repetition has no influence on the outcome of the \((k+1)\) th one, \(k=1,2, \ldots, m-1\). Show that the probability of the event

\[\left\{x_{1} \text { occurs } n_{1} \text { times, } x_{2} \text { occurs } n_{2} \text { times, } \cdots, x_{m} \text { occurs } n_{m} \text { times }\right\}\]

is given by

\[\frac{n !}{n_{1} ! n_{2} ! \cdots n_{m} !} p_{1}^{n_{1}} p_{2}^{n_{2}} \cdots p_{m}^{n_{m}} \quad \text { with } \quad \sum_{k=1}^{m} n_{k}=1\]

This probability distribution is called the multinomial distribution. It contains as a special case the binomial distribution \((n=2)\).