Question

A fair coin is tossed r times. Let Y be the number of heads in these r tosses. Assuming Y=y, we generate a Poisson random variable X with mean y. Find the variance of X. (Answer should be based on r).

Answer 1

P(head)=P=0.5

Y is number of heads in "r" tosses so

Y is Bin(r,0.5)

So E(Y)=r*p=0.5r

Var(Y)=r*p*(1-p)=r*0.5*(1-0.5)=0.25r

X|y~poisson (y)

So E(X|y)=y ,var(X|y)=y

We have to find Variance of X

Now

Var(X)=E(var(x|y))+var(E(x|y))

=E(y)+var(y)

=0.5r+0.25r

=0.75r

Hence answer is 0.75r