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KAUST
AMCS241/CS241/EE241 Probability and Random Processes

(Fall’13)
Homework 4

Pairs of Random Variables

Date: Tuesday 24th September, 2013.
Recommended Reading from Textbook: Chapter 5.1-5.8, 6.1-6.3.
Homeworks: Homework 4 e-mailed to students on Tuesday 24th September, 2013. Due in class
on Sunday 29th September, 2013 at 9:00 AM.

Problem 1:
Let FX(x) and FY (y) be valid one-dimensional CDF’s. Show that FX,Y (x,y) = FX(x)FY (y)
satisfies the properties of a two-dimensional CDF.

Problem 2:
Let X and Y be independent random variables each with distribution U(0,1). Let U = min{X,Y}
and V = max{X,Y}. Find E[U], and hence calculate cov(U,V ).

Problem 3:
The joint density function of two random variables X and Y is

fXY (x,y) =

{
kxy, 1 < x < 3 and 1 < y < 2

0, otherwise.

(a) What is the probability that X + Y < 3? (b) Are X and Y independent?

Problem 4:
Let X1 and X2 be independent random variables each having a uniform distribution on (0,1). A
stick of unit length is broken at points X1 and X2 from one of the ends. What is the probability
that the three pieces may be used to form a triangle?

Problem 5:
In this problem, we estimate π using probability arguments. Let X and Y be independent random
variables each with distribution U(−1,1). First relate P [X2 + Y 2 ≤ 1] to the value of π. Then
generate realizations of X and Y through a computer to estimate the value of π.