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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.