Suppose that tosses of a biased coin which come sup heads with probability 1/4 are independent. The coin is tossed 40 times and the number of heads X is counted. The coin is then tossed X more times.
a) Determine the expected total number of heads.
b) Determine the variance of the total number of heads.
My solution so far:
Let X1 be the total number of heads after the 40 first tosses. Let X2 be the total number of heads in the next set of tosses.
a) X1 ~ bin(1/4, 40) and X2 ~bin(1/4, x1)
E(X1 + X2) = E(X1) + E(X2) = 10 + 2.5 = 12.5
b) I am stuck here. If I use
var(X1+X2) = var(X1) + var(X2) +2cov(X1,X2)
How can I get cov(X1,X2)?
Cov[X,Y] = E[X1X2]-E[X1]E[X2] But what is E[X1X2]?
If I made a mistake in the first part please let me know about that too.