ECE 361 Homework 4 โ Solve any 5 of the 6 problems for FRIDAY FEBRUARY 7
1.) Random variables X and Y have the joint PMF
๐๐,๐(๐ฅ,๐ฆ)= {๐(๐ฅ2+ ๐ฆ2)if ๐ฅ โ {1,2,4} and ๐ฆ โ {1,3}
0 otherwise
a.) What is the value of the constant c?
b.) What is ๐(๐ < ๐)?
c.) What is ๐(๐ > ๐)?
d.) What is ๐(๐ = ๐)?
e.) What is ๐(๐ = 3)?
f.) Find the marginal PMFs ๐๐(๐ฅ) and ๐๐(๐ฆ)
g.) Find the expectations ๐[๐], ๐[๐], ๐[๐๐]
h.) Find the covariances var(๐), var(๐), var(๐ + ๐)
i.) Let A denote the event ๐ โฅ ๐. Find ๐[๐|๐ด] and var(๐|๐ด)
2.) Suppose that X, Y, and Z are independent random variables such that each is equal to 0 with
probability 0.5 and 1 with probability 0.5
a.) Compute the conditional probability ๐[๐ + ๐+ ๐ = 1|๐ โ ๐ = 0]
b.) Are the events {๐ = ๐} and {๐ = ๐} and {๐ = ๐} independent? Are they pairwise
independent? Explain.
3.) Jill sends her resume to 1000 companies she finds on monster.com. Each company responds
with probability 3/1000 (independent of what all other companies do). Let R be the number
of companies that respond
a.) Compute ๐[๐
]
b.) Computer var(๐
)
c.) Use a Poisson random variable approximation to estimate the probability ๐[๐
= 3]
4.) The number of calls arriving at a Police Station follows a Poisson distribution with rate
4.6/hour.
a.) What is the probability that exactly six calls will come between 8:00 PM and 9:00 PM?
b.) Find the probability that exactly seven calls will come between 9:00 PM and 10:30 PM.
