Advanced High-School Mathematics

SECTION 6.1 Discrete Random Variables 345

PROGRAM: PRIZES

:Input “NO OF PRIZES: ”, M

:Input “NO OF TRIALS: ”, N

:0→A

:For(I,1,N)

:For(L,1,M)

:0→L 2 (L)

:End

1 →B

:For(J,1,M)

:B*L 2 (J)→B

:End

:0→C

:While B< 1

:C+1→C

:randInt(1,M)→D

:L 2 (D)+1→L 2 (D)

:1→B

:For(J,1,M)

:B*L 2 (J)→B

:End

:End

:C→L 3 (I)

:End

:Stop

19. LetXbe the binomial random variable with success probabilityp
    and whereXmeasures the number of successes inntrials. Define
    the new random variable Y by settingY = 2X−n. Show that
    Y =y, −n≤y≤n can be interpreted as the total earnings after
    ngames, where in each game we win $1 with each success and we
    lose $1 with each failure. Compute the mean and variance ofY.


20. Continuing with the random variable Y given above, let T be
    the random variable which measures the number of trials needed
    to first observeY = 1. In other words,T is the number of trials
    needed in order to first observe one’s cumulative earnings reach $1.
    ThereforeP(T = 1) =p, P(T = 2) = 0, P(T = 3) =p^2 (1−p).
    Show that P(T = 2k+ 1) = C(k)pk+1(1−p)k, where C(k) =
    1
    k+ 1

Ñ

2 k

k

é

, n= 0, 1 , 2 ,...,are theCatalan numbers.

21. We continue the thread of Exercise 20, above. Show that ifp =
    1 /2—so the game is fair—then the expected time to first earn $1
    is infinite! We’ll outline two approaches here: a short (clever?)
    approach and a more direct approach.

(a) LetE be the expected waiting time and use a tree diagram as

on page 333 to show thatE =^12 +^12 (2E+ 1), which implies