ECE2191 Probability Models in Engineering
Tutorial 4: Discrete Random Variables
Second semester 2021
1. Let X be a discrete random variable with range RX = {1, 2, 3, ...}. Suppose the PMF of X is given by
(a) Find and plot the CDF of X, FX(x).
(b) Find P (2 < X ≤ 5).
(c) Find P (X > 4).
2. Let X be a random variable with PMF
(a) Find the a and E[X].
(b) What is the PMF of the random variable Z = (X - E[X])2?
(c) Using part b, compute the variance of X.
(d) Compute the variance of X using the formula VAR[X] = Σx(X - E[X])2 PX(x).
3. You just rented a large house and the realtor gave you 5 keys, one for each of the 5 doors of the house. Unfortunately, all keys look identical, so to open the front door, you try them at random.
(a) Find the PMF of the number of trials you will need to open the door, under the following alternative assumptions: (1) after an unsuccessful trial, you mark the cor- responding key, so that you never try it again, and (2) at each trial you are equally likely to choose any key.
(b) Repeat part 3a for the case where the realtor gave you an extra duplicate key for each of the 5 doors.
4. A family has 5 natural children and has adopted 2 girls. Each natural child has equal probability of being a girl or a boy, independently of the other children. Find the PMF of the number of girls out of the 7 children.
5. Let X be a random variable that takes values from 0 to 9 with equal probability 1/10.
(a) Find the PMF of the random variable Y = X mod (3).
(b) Find the PMF of the random variable Y = 5 mod ( X + 1)
6. Fischer and Spassky play a chess match in which the first player to win a game wins the match. After 10 successive draws, the match is declared drawn. Each game is won by Fischer with probability 0.4, is won by Spassky with probability 0.3, and is a draw with probability 0.3, independently of previous games.
(a) What is the probability that Fischer wins the match?
(b) What is the PMF of the duration of the match?
7. Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) P (X = 0) and (b) P (X > 2)
8. When coin 1 is flipped, it lands on heads with probability 0.4; when coin 2 is flipped, it lands on heads with probability 0.7. One of these coins is randomly chosen and flipped 10 times.
(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
(b) Given that the first of these ten flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?
9. A city’s temperature is modeled as a random variable with mean and standard deviation both equal to 10 degrees Celsius. A day is described as normal if the temperature dur- ing that day ranges within one standard deviation from the mean. What would be the temperature range for a normal day if temperature were expressed in degrees Fahrenheit?
10. As an advertising campaign, a chocolate factory places golden tickets in some of its candy bars, with the promise that a golden ticket is worth a trip through the chocolate factory, and all the chocolate you can eat for life. If the probability of finding a golden ticket is p, find the mean and the variance of the number of candy bars you need to eat in order to find a ticket.
11. [Optional] The number of customers arriving at Mr. Kim’s convenience store is a Poisson random variable. On average 10 customers arrive per hour. Let X be the number of customers arriving from 10am to 11:30am. What is P (10 < X ≤ 15)?
12. [Optional] You go to a party with 500 guests. What is the probability that exactly one other guest has the same birthday as you? Calculate this exactly and also approximately by using the Poisson PMF. (For simplicity, exclude birthdays on February 29.)
13. [Optional] Let K be a random variable that takes, with equal probability 1/(2n + 1), the integer values in the interval [-n, n]. Find the PMF of the random variable Y = ln X , where X = a|K|, and a is a positive number