Reading and Problem Assignment #5
Math 501ā1, Spring 2006
University of Utah
Read Chapter 4 (expectations). Start reading Chapter 5 (continuous random vari-
ables).
The following are borrowed from your text.
Problems:
1. A gambling book recommends the following āwinnin strategyā for the game of roulette:
It recommends that the gambler bet $1 on red. If red appears (this has probability
18/38), then the gambler should take her $1 profit and quit. Else, she should make
additional $1 bets on red on each of the next two spins of the roulette wheel, and then
quit. Let Xdenote the gamblerās winnings when she quits.
(a) Find P{X > 0}.
(b) Are you convinced that the strategy is indeed a āwinningā strategy? Explain
your answer.
(c) Compute EX .
2. One of the numbers 1 through 10 is chosen at random. You are to try and guess the
number chosen by asking questions with āyes-noā answers. Compute the expected
number of questions that you need to ask in each of the following two cases:
(a) Your ith question is to be, āIs it i?ā, for i= 1, . . . , 10.
(b) With each question you try to eliminate one-half of the remaining numbers, as
nearly as possible. [E.g., āIs it greater than or equal to 5?ā, etc.]
3. A person tosses a fair coin until a tail appears for the first time. If ātailā appears on
the nth flip, then the person wins 2ndollars. Let Xdenote the personās winnings.
Show that EX =ā. This is known as the āSt.-Petersbourg paradox.ā
4. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If
they are the same color, then you win $1.10; if they are different colors, then you win
ā$1.00 (i.e., you lose one dollar). Compute:
(a) the expected value of the amount you win;
(b) the variance of the amount you win.
Theoretical Problems:
1. Let Xbe such that P{X= 1}=pand P{X=ā1}= 1 āp. Find a constant c6= 1
such that E[cX] = 1.
2. Prove that if Xis a Poisson random variable with parameter Ī», then for all nā„1,
E[Xn] = Ī»E[(X+ 1)nā1].
Use this to compute EX ,E[X2], VarX, and E[X3].