## Description

Exercise 1.

Two dice are tossed. Let X be the absolute difference in the number of dots facing up. Let

g(X) = (

X2

, if X > 2

0, otherwise.

and h(X) = −|X − 2|.

(a) Find E[g(X)].

(b) Find E[h(X)].

c 2020 Stanley Chan. All Rights Reserved. 1

Exercise 2.

A modem transmits a +2 voltage signal into a channel. The channel adds to this signal a noise term that is

drawn from the set {0, −1, −2, −3} with respective probabilities {1/10, 2/10, 3/10, 4/10}.

(a) Find and sketch the PMF and the CDF of the output Y of the channel.

(b) What is the probability that the output of the channel is equal to the input of the channel?

(c) What is the probability that the output of the channel is positive?

(d) Find the expected value and variance of Y .

c 2020 Stanley Chan. All Rights Reserved. 2

Exercise 3.

A voltage X is uniformly distributed in the set {0, 1, 2, 3}.

(a) Find the mean and variance of X.

(b) Find the mean and variance of Y = X2 − 2.

(c) Find the mean of W = sin(πX/4).

(d) Find the mean of Z = sin2

(πX/4).

c 2020 Stanley Chan. All Rights Reserved. 3

Exercise 4.

(a) If X is Poisson(λ), compute E[3/(X + 2)].

(b) If X is Bernoulli(p) and Y is Bernoulli(q), compute E[(2X + Y )

2

] if X and Y are independent.

c 2020 Stanley Chan. All Rights Reserved. 4

Exercise 5.

Let X be the number of photons counted by a receiver in an optical communication system. It is known

that X is a Poisson random variable with rate λ1 when a signal is present and a Poisson random variable

with rate λ0 < λ1 when a signal is absent. The probability that the signal is present is p. Suppose that we

observe X = k photons. We want to determine a threshold T such that if k ≥ T then we claim that the

signal is present; and if k < T then we claim that the signal is absent. What is this T?

c 2020 Stanley Chan. All Rights Reserved. 5

Exercise 6.

A random variable X has PDF:

fX(x) = (

cx(4 − x

2

), 0 ≤ x ≤ 2

0, otherwise.

(a) Find c

(b) Find FX(x)

(c) Find E[X]

c 2020 Stanley Chan. All Rights Reserved. 6

Exercise 7.

Consider a CDF

FX(x) =

0, if x < −2

0.25, if − 2 ≤ x < 0

(x + 1)/2, if 0 ≤ x < 1

1, otherwise.

(a) Find P[X < −2], P[X = 0] and P[X > 0.5].

(b) Find fX(x).

c 2020 Stanley Chan. All Rights Reserved. 7

Exercise 8.

Consider a discrete PMF pX(k) =

0.3 0.1 0.15 0.25 0.1 0.08 0.02

. Write a MATLAB / Python

function that takes this PMF and generates N = 100, 000 realizations of X. Your function can only use

the uniform random number generator rand in MATLAB (or numpy.random.rand in Python) and no other

random number generators. Submit your code and the empirical histogram of X.

Please attach your code and plot after this page.

c 2020 Stanley Chan. All Rights Reserved. 8