Let X denote the number of eggs laid by a bird Suppose that

Let X denote the number of eggs laid by a bird. Suppose that P(X = x) = [(4^x)e^(4)]/x! for x = 0, 1, 2, · · · .
Assume that each Suppose that each egg has a 20% chance of being successfully hatched to become a
bird. Let Y denote the total number of successfully hatached eggs.
(a) What is the distribution of Y conditional on X = x?
(b) Find P(X = 3, Y = 2).

(c)In the problem above, find the expected number eggs that are laid by a bird.

Solution

Here X~poisson with 4 then P(X=x)=(4^x*e^-4)/(x!) and Y/X~B(x,p),then F_y/x(y/x)=P[Y<=y/X=x]=(xCx)(.2)^x(4^x*e^-4)/(x!). P(X=3,Y=2)=P(X=3)*P(Y=2)=((4³(e^-4))/3!)*((3c2)(.2)²(.8)¹)=.0187, E(Y)=np=3(.2)=.6, i.e., 6% of eggs are laid by a bird.