Question

Let X be a Bin(n, p) random variable. Show that Var(X) = np(1 − p). Hint: First compute E[X(X − 1)] and then use (c) and (d).

(c) Var(X) = E(X^2 ) − (E X)^ 2

(d) E(X + Y ) = E X + E Y

Answer 1

X is Binomial(n,p). The probability mass function (pmf) of X is

First we calculate the expectation of X

Now the term in the summation corresponds to binomial probability for Y distributed Binomial(n-1,p) with pmf

Since the pmf sums to 1, the summation is equal to 1, that is

Hence

Next we will compute the expected value of E[X(X-1)]

Now the term in the summation corresponds to binomial probability for Y distributed Binomial(n-2,p) with pmf

Since the pmf sums to 1, the summation is equal to 1

That means

But

Now to the variance of X