Chapter 5.3, Problem 19E

Flip $n=8$ fair coins and remove all that came up heads. Flip the remaining coins (that came up tails) and remove the heads again. Continue flipping the remaining coins until each has come up heads. We shall find the pmf of Y, the number of trials needed. Let $X_i$ equal the number of flips required to observe heads on coin $i,i=1,2,\mathrm{..}.,8$.

Then $Y=\max \left(X_1,X_2,\mathrm{...},X_8\right)$.

(a) Show that $P\left(Y\le y\right)={\left[1-{\left(\frac{1}{2}\right)}^y\right]}^8$.

(b) Show that $P\left(Y=y\right)={\left[1-{\left(\frac{1}{2}\right)}^y\right]}^8-{\left[1-{\left(\frac{1}{2}\right)}^{y-1}\right]}^8,y=1,2,\mathrm{...}$

(c) Use a computer algebra system such as Maple or Mathematica to show that E(Y) = 13.315424/ 3,011,805 = 4.421.

(d) What happens to the expected value of Y as the number of coins is doubled?