EECS 501__________________________PROBLEM SET #5__________________________Fall 2001

ASSIGNED: October 12, 2001. Read Stark and Woods pp. 178-219. THIS WEEK:
DUE DATE: October 19, 2001. Moments, inequalities, and characteristic functions.
1. Stark and Woods #4.12. First determine E[y|Ø], then multiply by pdf and integrate.
2. Stark and Woods #4.14. fz(Z) is a Rayleigh pdf. Use integration by parts.
3. Iterated expectation. Let x and y be two random variables (RVs) with joint pdf fx,y(X,Y).
Let g(x,y) be any function of x and y, and Ex[·] denote expectation wrt x alone.
1. Show E[g(x,y)]=Ey[Ex[g(x,y)|y]]. In words ("wrt" means "with respect to"):
Can compute expectation wrt x, holding y constant, and then compute expectation wrt y.
2. Use this to show E[x]=Ey[E[x|y]] and Var[x]=Ey[Var[x|y]]+Vary[E[x|y]].

4. Use the result of the previous problem to solve this one.
Let y=x1+...+xn where xi are independent and identically distributed random variables
and n is itself a discrete random variable. Show that:
1. E[y]=E[n]E[x] and Var[y]=E[n]Var[x]+Var[n]E[x]2 (note the asymmetry).
• Use these formulae to solve the following problem:
• The number k of customers in a market has a Poisson pmf with parameter µ1.
• Number n of items purchased by any customer has a Poisson pmf with parameter µ2.
To increase the expectation of total number of items purchased by 10%, the store can either:
(a) increase µ2 by 10%; or (b) increase µ1 by 10%. Assume n and k are independent.
Which of (a) and (b) leads to the smaller variance of total number of items purchased?

5. Order statistics. Let x1,x2...xn be n independent RVs, each having pdf fx(X) and PDF Fx(X).
Order the x1...xn into y1 < y2 < ... < yn as follows: Let y1=smallest of {x1...xn};
y2=2nd smallest of {x1...xn}; and so on up to yn=largest of {x1...xn}.
1. Show that the joint pdf fy1...yn(Y1...Yn) =n!fx(Y1)fx(Y2)...fx(Yn) for Y1 < Y2 < ... < Yn.
2. Compute the pdf fyk(Yk) for the kth smallest by integrating this over all the other Yi.
HINT: This is similar to Problem Set #3, Problem #5.
3. Determine fyk(Yk) directly using fyk(Y)= Pr[Y < yk < Y+dY]
=(some #ways)Pr[x < Y]k-1Pr[Y < x < Y+dY] Pr[x > Y]n-k.

6. Let x be a random variable with E[x]=0. If x > 0 or x=0, prove that Pr[x=0]=1.
HINT: Let An={w: x(w) > 1/n} for w in the sample space.
Show 0 < Pr[An] < nE[x] and use continuity of probability.
"An elephant is a mouse built to government specifications."