ECE2191 Probability Models in Engineering
Tutorial 7: Pairs of Random Variables
Second Semester 2021
1. A stock market trader buys 100 shares of stock A and 200 shares of stock B . Let X and Y be the price changes of A and B , respectively, over a certain time period, and assume that the joint PMF of X and Y is uniform over the set of integers x and y satisfying
-2 ≤ x ≤ 4, - 1 ≤ y - x ≤ 1
(a) Find the marginal PMFs and the means of X and Y.
(b) Find the mean of the trader's profit.
2. A class of n students takes a test consisting of m questions. Suppose that student i submitted answers to the first mi questions.
(a) The grader randomly picks one answer, call it (I, J), where I is the student ID number (taking values 1, ..., n) and J is the question number (taking values 1, ..., m). Assume that all answers are equally likely to be picked. Calculate the joint and the marginal PMFs of I and J.
(b) Assume that an answer to question j, if submitted by student i, is correct with prob- ability pij . Each answer gets “a” points if it is correct and gets “b” points otherwise. Calculate the expected value of the score of student i.
3. On a given day, your golf score takes values from the range 101 to 110, with probability 0.1, independently from other days. Determined to improve your score, you decide to play on three different days and declare as your score the minimum X of the scores X1 , X2 , and X3 on the different days.
(a) Calculate the PMF of X .
(b) By how much has your expected score improved as a result of playing on three days?
4. Consider four independent rolls of a 6-sided die. Let X be the number of 1's and let Y be the number of 2's obtained. What is the joint PMF of X and Y?
5. Alice passes through four traffic lights on her way to work, and each light is equally likely to be green or red, independently of the others.
(a) What is the P MF , the mean, and the variance of the number of red lights that Alice encounters?
(b) Suppose that each red light delays Alice by exactly two minutes. What is the variance of Alice’s commuting time?
6. Sam will read either one chapter of his probability book or one chapter of his history book. If the number of misprints in a chapter of his probability book is Poisson distributed with mean 2 and if the number of misprints in his history chapter is Poisson distributed with mean 5, then assuming Sam is equally likely to choose either book, what is the expected number of misprints that Sam will come across?
7. Consider two random variables X and Y with joint PMF given in the figure below. Let Z = E[X|Y].
(a) Find the marginal PMFs of X and Y.
(b) Find the conditional PMF of X given Y = 0 and Y = 1; i.e., find PX|Y(x|0) and PX|Y (x|1).
(c) Find the PMF of Z.
(d) Find E[Z], and check that E[Z] = E[X].
(e) Find Var(Z).
8. A miner is trapped in a mine containing three doors. The first door leads to a tunnel that takes him to safety after two hours of travel. The second door leads to a tunnel that returns him to the mine after three hours of travel. The third door leads to a tunnel that returns him to his mine after five hours. Assuming that the miner is at all times equally likely to choose any one of the doors, what is the expected length of time until the miner reaches safety?
9. Assume that X and Y have the following joint pdf
(a) Find c and the joint cdf FX;Y(x, y).
(b) Find the marginal cdf’s, FX and Fy , and the marginal pdf’s, fX and fY .
(c) Find E[X] and Var[X].
(d) Find the correlation, covariance, and correlation coefficient of X and Y.
10. A new virus, called COVID-20, has recently spread out in a land called Wonderland. This new coronavirus has a strange infection behaviour with respect to gender. Assuming that X and Y represent the portion of the males and females who have been infected by the virus, respectively, scientists derived the joint pdf of X and Y as
Determine the probability that the portion of the infected males is less than or equal to 0.3.
11. [Optional] Let X and Y be two independent Poisson random variables with parameters αx and αy . Show that Z = X + Y is also a Poisson random variable. What about the sum of n independent Poisson random variables: is it Poisson as well?
12. [Optional] Suppose that the number of customers visiting a fast food restaurant in a given day is N , a Poisson random variable with parameter α . Assume each customer purchases a drink with probability p, independently from other customers, and independently from the value of N. Let X be the number of customers who purchase drinks. Let Y be the number of customers that do not purchase drinks, so X + Y = N.
(a) Find the marginal PMFs of X and Y.
(b) Find the joint PMF of X and Y.