GEORGIA INSTITUTE OF TECHNOLOGYSchool of Electrical and Computer Engineering

ECE 6273

Problem Set #0

Reading: Review your probability and linear algebra.

Date assigned: August 23, 2010Date due: Not collected

This is a diagnostic problem set that is intended to determine whether or not your back-ground in basic probability theory is sufficient to meet the course prerequisite. It will notbe collected or graded, but a solution set will be provided. If you have trouble with anyproblem, do not hesitate to look for help.

Problem 0.1

A random variable X has probability distribution function

PX(x) =

0, x < −100.1(x + 10), −10 ≤ x < −70.3, −7 ≤ x < −51− 0.3 exp[−(x + 5)], −5 ≤ x

(a) Calculate the following probabilities:

Pr[X ≤ −9], Pr[X = 1], Pr[X < −5], Pr[X ≥ 0]

(b) Find the probability density function, pX(x), of the random variable X.

(c) Find the conditional probability density function pX|A(x|A) where A is the event X > 0.

Problem 0.2:

Let X and Y be statistically independent random variables with probability densityfunctions

pX(x) =1

2δ(x + 1) +

1

2δ(x− 1)

and pY (y) = exp(−y2/2σ2)/(2πσ2)1/2, and let Z = X + Y , and W = XY .

(a) Find the probability density function of Z, pZ(z), and the probability density functionof W, pW (w).

(b) Find the conditional probability density functions pZ|X(z|x = −1) and pZ|X(z|x = 1).

(c) Find the mean values Y , Y , the variances, σ2Y , σ2

W , and the covariance λY W . Are Y andW uncorrelated random variables? Are Y and W statistically independent randomvariables?

Problem 0.3:

Let X and Y be random variables with joint density

pX,Y (x, y) =

{2 0 < x < y < 10 elsewhere

(a) Find the marginal densities pX(x), pY (y).

(b) Let Z = X + Y , find pZ(z).

Problem 0.4:

Let X be a random variable with characteristic function

ΦX(jω) =λ2

λ2 + ω2

Find X and σ2X .

Problem 0.5:

A dart is thrown at random at a wall. Let (X, Y ) denote the Cartesian coordinates of thepoint in the wall pierced by the dart. Suppose that X and Y are statistically independentGaussian random variables, each with mean zero and variance σ2, i.e.,

pX(α) = pY (α) = (2πσ2)−1/2 exp(−α2/2σ2)

(a) Find the probability that the dart will fall within the σ-radius circle centered on thepoint (0,0).

(b) Find the probability that the dart will hit in the first quadrant (X ≥ 0, Y ≥ 0).

(c) Find the conditional probability that the dart will fall within the σ-radius circle centeredon (0,0) given that the dart hits in the first quadrant.

(d) Let R = (X2 + Y 2)1/2, and Θ = tan−1(Y/X) be the polar coordinates associated with(X, Y ). Find Pr[0 ≤ R ≤ r, 0 ≤ Θ ≤ θ] and obtain pR,Θ(r, θ).