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MATH/STATS 525
Homework 5 (due Thursday, October 13)
Problem 1. Show that Var(a + X) = Var(X) for any random variable X and constant a.
Problem 2.
(a) Show that for a discrete random variable X, if E(X^2 ) = 0, then X = 0 almost surely (i.e. P(X = 0) = 1).
(b) Show that a discrete random variable X, if Var(X) = 0, then X is almost surely constant (i.e. there exists c ∈ R such that P(X = c) = 1).
Problem 3. Let Ω = {w 1 , w 2 , w 3 } be the sample space. The probability measure is given by P(w 1 ) = P(w 2 ) = P(w 3 ) = 13. Define random variables X, Y : Ω → R by
X(w 1 ) = 1, X(w 2 ) = 2, X(w 3 ) = 3, Y (w 1 ) = 2, Y (w 2 ) = 3, Y (w 3 ) = 1.
(a) Show that X and Y have the same mass function.
(b) Find the mass functions of X + Y and XY.
Problem 4. A total of n bar of magnets are placed end to end in a line with random independent orientations. Adjacent like poles repel while ends with opposite polarities join to form blocks. Let X be the number of blocks of joined magnets. Find E(X) and Var(X). (Hint: Note that X = 1 + (I 1 + I 2 + · · · + In− 1 ) where Ii is the indicator function that ith magnet and i + 1th magnet have the opposite orientation.)
Problem 5. A secretary drops n (n ≥ 2) matching pairs of letters and envelopes down the stairs, and then places the letters into the envelopes in a random order. Let X be the number of correctly matched pairs. Compute E(X) and Var(X). (Hint: Use the indicator function Ij that the jth letter is placed into the matching envelope as in the example in the class. However, you do not need to compute the mass function of X as we did in the class. It is enough to compute E(Ij ) and E(Ij Ik) for j 6 = k.)
Problem 6. You roll a fair die repeatedly. If it shows 1, then you must stop. But you may stop at any prior time. Your score is the number shown by the die on the final roll. Now you would like to find a stopping strategy that yields the greatest expected score. Consider the following strategy: you stop the first time that the die shows r or greater (therefore, you stop the first time that the die shows either 1 or r or r + 1, or... , or 6). Let S(r) be the expected score achieved by following this strategy. Compute S(r) for r = 2, 3 ,... , 6 and determine which of them is the greatest. (Hint: Note that S(6) = 6 · P(6 occurs before 1) + 1 · P(1 occurs before 6).)
Problem 7. Let X and Y have joint mass function
f (j, k) =
c(j + k)aj+k j!k!
, j, k = 0, 1 , 2 ,... ,
where a is a positive constant. Find c, P(X = j), E(X) and P(X + Y = r).
Problem 8. (Bernoulli’s diffusion model) Let n ≥ 2. Urn R contains n red balls and urn B contains n blue balls. At each stage, a ball is selected at random from each urn, and they are swapped. We will compute the mean number of red balls in urn R after k stage. Label each of n red balls by 1, 2 ,... , n. Let Ai be the event that the ith red ball is in urn R after k stages. Then the number Nk of red balls in urn R after k stages satisfies Nk = IA 1 + IA 2 + · · · + IAn.
(a) The ith red ball is in urn R after k stages if and only if it is swapped even number of times. Using this show that
P(Ai) =
∑^ k
m=0;m even
k m
n
)m( 1 −
n
)k−m .
(b) Simplify the formula in (a) using the identity
1 2
[(y + x)k^ + (y − x)k] =
∑^ k
m=0;m even
k m
xmyk−m
(this follows from the binomial identity; prove it by yourself), and then show that
E(Nk) =
n 2
n
)k^ ) .
