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."