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.
- Stark and Woods #4.12. First determine E[y|Ø], then multiply by pdf and integrate.
- Stark and Woods #4.14. f_{z}(Z) is a Rayleigh pdf. Use integration by parts.
- 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.
- 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.
- Use this to show E[x]=E_{y}[E[x|y]] and Var[x]=E_{y}[Var[x|y]]+Var_{y}[E[x|y]].
- Use the result of the previous problem to solve this one.
Let y=x_{1}+...+x_{n} where x_{i} are independent and identically distributed
random variables
and n is itself a discrete random variable. Show that:
- 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 µ_{1}.
- Number n of items purchased by any customer has a Poisson pmf with parameter µ_{2}.
To increase the expectation of total number of items purchased by 10%, the store can either:
(a) increase µ_{2} by 10%; or (b) increase µ_{1} by 10%. Assume n and k
are independent.
Which of (a) and (b) leads to the smaller variance of total number of items purchased?
- Order statistics. Let x_{1},x_{2}...x_{n} be n independent RVs,
each having pdf f_{x}(X) and PDF F_{x}(X).
Order the x_{1}...x_{n} into y_{1} < y_{2} < ... < y_{n}
as follows: Let y_{1}=smallest of {x_{1}...x_{n}};
y_{2}=2^{nd} smallest of {x_{1}...x_{n}};
and so on up to y_{n}=largest of {x_{1}...x_{n}}.
- Show that the joint pdf f_{y1...yn}(Y_{1}...Y_{n})
=n!f_{x}(Y_{1})f_{x}(Y_{2})...f_{x}(Y_{n}) for
Y_{1} < Y_{2} < ... < Y_{n}.
- 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.
- Determine f_{yk}(Y_{k}) directly using f_{yk}(Y)=
Pr[Y < y_{k} < Y+dY]
=(some #ways)Pr[x < Y]^{k-1}Pr[Y < x < Y+dY]
Pr[x > Y]^{n-k}.
- 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.
"An elephant is a mouse built to government specifications."