Show that if Xi follows a binomial distribution with trials

Show that if X_i follows a binomial distribution with trials n and probability of success p_i=p, i=1,…n and X_i are independent then   ⁿ_i=1 X_i follows binomial distribution.

Solution

Let Xi=1 if the ith Bernoulli trial is successful or o otherwise.

Then, X=summationXi, where Xi’s are independent and identically distributed.

Where X=number of successes

Hence X follows a binomial distribution with mean np and variance npq with parameters n and p.