Question

Complete the following:

(a) Var (X) ≥ 0 for any discrete r.v. X. Also, explain why X is constant if and only if

Var (X) = 0. Hint: write Var (X) = E (X − µ)2 = Px (x − µ)2 pX (x), where µ = E (X).

(b) Let X represent the dierence between the number of heads and the number of tails obtained

when a fair coin is tossed n times. Find E (X) and Var (X)

Answer 1

a) Var (X) =0

=> E (X − µ)2 = 0

=> X = µ ie. there is only one value that X takes ie. µ in this case. Hence, the distribution if degenerate since the variation among the variables is 0 which is only possible if the random variable takes only one value.

Now if X is a constant then the mean is also the same constant.

As a result X- µ = 0 , therefore as a resultant of this Var(X) = 0.

b) X : no.of heads - no.of tails = Y - Z (say)

Also Y+Z = n

Here, Y~binomial (n , p) and Z ~binomial( n, q)

E(X) = E( Y-Z) = E(Y) - E(Z) = np - nq = n(p-q)

Var(X) = Var( Y-Z) = Var(Y) + Var(Z) - 2cov (Y,Z) = npq + npq - 2cov( Y , n - Y) = npq+npq+ 2npq = 4npq