The moment generating function of X is given by

M_X(t) =e^tp/1(1p)e^t

Now let X₁ be the number of tests up to and including the first positive, X₂ the number of tests from the first positive up to and including the second positive, and further X_r the number of tests from (r 1) positive up to and including r positive. Then Y = ^r_i=1 Xi number of tests up to and including r positive test.

c) Find E (Y) expectation and V (Y) variance and calculate the expected number of tests performed up to and including the 10th positive, if p is still 0.001.

d) Find expression for P(Y=k), k = r, r + 1, ....