Solved STA219: Probability and Statistics for Engineering Assignment 4

Description

1. (10 points) Suppose the PDF of random variable Y is
𝑓𝑌
(𝑦) = {
5𝑦
4
, 0 < 𝑦 < 1,
0, otherwise.
Also, the conditional PDF of 𝑋 given 𝑌 = 𝑦 is
𝑓𝑋|𝑌
(𝑥|𝑦) = {
3𝑥
2
𝑦
3
, 0 < 𝑥 < 𝑦 < 1,
0, otherwise.
What is the probability 𝑃(𝑋 > 0.5)?
2. (10 points) Suppose 𝑋 and 𝑌 are independent discrete random variables, and the joint PMF of
(𝑋, 𝑌) is as follows:
X\Y 𝑦1 𝑦2 𝑦3
𝑥1 𝑎 1/9 𝑐
𝑥2 1/9 𝑏 1/3
Please calculate the values of 𝑎, 𝑏, 𝑐.
3. (10 points) Let 𝑋 and 𝑌 be independent Poisson random variables with parameter 𝜆 , i.e.,
𝑋~Poisson(𝜆), 𝑌~Poisson(𝜆). Let 𝑈 = 2𝑋 + 𝑌, 𝑉 = 2𝑋 − 𝑌. Please calculate the correlation
coefficient between 𝑈 and 𝑉.
4. (10 points) Toss a coin 𝑛 times, with 𝑋 and 𝑌 representing the number of heads and tails,
respectively. What is the covariance and correlation coefficient between 𝑋 and 𝑌?
5. (15 points) Suppose the joint PDF of random vector (𝑋,𝑌) is
𝑓(𝑥, 𝑦) = {
1, |𝑦| < 𝑥, 0 < 𝑥 < 1,
0, otherwise.
(1) Please calculate E(𝑋), E(𝑌), Cov(𝑋, 𝑌). (10 points)
(2) Is 𝑋 and 𝑌 independent? (5 points)

6. (10 points) Let 𝑋 and 𝑌 be independent Uniform random variables, i.e., 𝑋~U(0,1), 𝑌~U(0,1).
What is the PDF of 𝑇 = 𝑋 + 𝑌?
7. (10 points) Let 𝑋1
, 𝑋2
and 𝑋3
represent the time (in minutes) necessary to perform three
successive repair tasks at a service facility. They are independent, normal random variables with
expected values 45, 50 and 75, and variances 10, 12 and 14, respectively. What is the probability that
the service facility can finish all three tasks within 3 hours (that is, 180 minutes)?
8. (10 points) There are 40 light bulbs, and the lifespan of each light bulb follows an exponential
distribution with an average lifespan of 25 days. Suppose that we use one light bulb at a time and
replace it immediately with a new bulb once the previous one breaks. Please find the probability that
these bulbs can be used for a total of more than 900 days.
9. (15 points) A large hotel has a total of 500 rooms, and each room has one air conditioner with a
power rating of 2 kW. Suppose the occupancy rate is 80%, which means that each room has an 80%
probability of being occupied, independently of other rooms. How many kW of power are needed to
ensure a 99% probability of having enough power for the air conditioners?