EECS 126: Probability and Random Processes
Midterm 2
Tuesday, November 12, 2002 3:40-5 pm
Answer all questions on the attached blank sheets. Please explain your reasoning carefully.
Answers without reasoning will get no marks. The total is 100 points. Q. 2 (d) is a bonus
part worth 15 points.
[20] 1. Let Xand Ybe independent RV’s, both exponentially distributed with mean 1.
Let Z= max(X, Y ).
[10] a) Compute the pdf of Xconditional on the event that Z≤z.
[10] b) Xand Yare independent, but conditional on Z=z, are they still independent?
Explain. (Hint: no calculations needed.)
[35] 2.
[10] a) Suppose you have access to a RV X∼N(µ1, σ2
1). Explain how you can use Xto
generate a RV Ywhich is N(µ2, σ2
2).
[10] b) Suppose you have access to a RV Xuniform distributed in [0,1]. Explain how you
can use Xto generate a continuous RV Ywith a given pdf fYwhich is nonzero everywhere.
[15] c) Suppose you have access to a RV Xuniform distributed in [0,1]. Explain how you
can use Xto generate a discrete RV Ywith a given pmf pY(·). You can assume that the
range of values Ycan take on is finite. (Hint: try the simple case when Yis Bernouilli
(i.e.Ytakes on only two possible values) and then attack the general case.)
[15] d) (Bonus) Suppose you have access to two indpendent RVs Xand Yboth uniform
distributed in [0,1]. Explain how you can use them to generate two continuous RV’s Uand
Vwith a given joint density fU,V which is nonzero everywhere. (Hint: you may want to
consider a sequential procedure where you first generate Uand then generate V.)
