EECS 501__________________________PROBLEM SET #5__________________________Fall 2001

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ASSIGNED: October 12, 2001. Read Stark and Woods pp. 178-219. THIS WEEK:
DUE DATE: October 19, 2001. Moments, inequalities, and characteristic functions.

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1. Stark and Woods #4.12. First determine E[y|Ø], then multiply by pdf and integrate.
2. Stark and Woods #4.14. f_z(Z) is a Rayleigh pdf. Use integration by parts.

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3. Iterated expectation. Let x and y be two random variables (RVs) with joint pdf f_x,y(X,Y).
   
   Let g(x,y) be any function of x and y, and E_x[·] denote expectation wrt x alone.
   1. Show E[g(x,y)]=E_y[E_x[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]=E_y[E[x|y]] and Var[x]=E_y[Var[x|y]]+Var_y[E[x|y]].

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4. Use the result of the previous problem to solve this one.
   
   Let y=x₁+...+x_n where x_i 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).
   2. 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?

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5. Order statistics. Let x₁,x₂...x_n be n independent RVs, each having pdf f_x(X) and PDF F_x(X).
   Order the x₁...x_n into y₁ < y₂ < ... < y_n as follows: Let y₁=smallest of {x₁...x_n};
   y₂=2^nd smallest of {x₁...x_n}; and so on up to y_n=largest of {x₁...x_n}.
   
   1. Show that the joint pdf f_y1...yn(Y₁...Y_n) =n!f_x(Y₁)f_x(Y₂)...f_x(Y_n) for Y₁ < Y₂ < ... < Y_n.
   2. Compute the pdf f_yk(Y_k) for the k^th smallest by integrating this over all the other Y_i.
      HINT: This is similar to Problem Set #3, Problem #5.
   3. Determine f_yk(Y_k) directly using f_yk(Y)= Pr[Y < y_k < Y+dY]
      =(some #ways)Pr[x < Y]^k-1Pr[Y < x < Y+dY] Pr[x > Y]^n-k.

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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 A_n={w: x(w) > 1/n} for w in the sample space.
   Show 0 < Pr[A_n] < nE[x] and use continuity of probability.

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   "An elephant is a mouse built to government specifications."