EE3025 - Statistical Methods in ECE
Spring 2024
Homework 3
Due on Friday, Sunday 18, 2024, 12:00 PM
Problem 1
(a) Tom, Jane, Sarah, and Sam are chatting, and they accidentally find that Tom and Sarah have the same birthday (day and month)! This is very unlikely, right?! Find the probability that there are people with the same birthday in a party of 4 persons.
Hint: Assume only 365 days for a year. First, find the probability that no two people have the same birthday. (b) Repeat part (a) for a group of 10 people.
(c) There are 60 students in the probability class. What is the chance that at least two people in the room have the same birthday? You can use any computational software (e.g., MATLAB) to evaluate this probability.
Problem 2
Let X be the number of cars being repaired at a repair shop. We have the following information:
• At any time, at most 3 cars are being repaired. The probability of having 2 cars at the shop is the same as the probability of having 1 cars.
• The probability of having no car at the shop is the same as the probability of having 3 cars.
• The probability of having 1 or 2 cars is half of the probability of having 0 or 3 cars.
(a) Find the probability mass function (p.m.f) of X.
(b) Determine the cumulative distribution function (c.d.f.) of X and draw it.
(c) Determine the mode and median of the random variable X.
(d) Compute the expected value and variance of X.
Problem 3
We have a deck of 52 cards. Each card is la- beled by a suit from {%, ◇ , A, ◇} and a rank from {2, 3, 4, . . . , 10,J,Q,K, A}. A poker hand consists of 5 cards.
(a) A two-pair hand is a hand that contains two cards of one rank, two cards of another rank and one card of a third rank, such as {J%, J◇, 4◇, 4%, 9◇}. What is the number of all possible two-pair hands?
(b) A four-of-a-kind hand is a hand that contains four cards of one rank and one card of another rank. For example {3%, 3◇, 3◇, 3A, A◇} is a four-of-a-kind hand. How many different four-of-a-kind hands exist?
(c) A full-house in poker is a hand where three cards share one rank and two cards share another rank. For example {3%, 3◇, J◇, J◇,JA} is a full house hand. Find the number of different full-house hands?
(d) Assume you have only seen 4 of your cards, which are {3%, 3◇, 3◇, 9%}. What is the probability that you have a full-house hand after receiving your fifth card?
(e) The dealer looks at your hand and tells you that it has at least two cards with identical rank. What is the probability that your hand is a four-of-a-kind?
Hint: First find the probability that a hand has at least two cards with identical rank.
Problem 4
There are 11 seats in the congress of a small country, each assigned to one region. Two major parties A and B are competing in the election. Let pbe the probability that party A’s nominee wins in each region.
(a) Let NA be the number of members of party A who are selected for the congress. What is the range of NA? What is the p.m.f. of NA?
(b) For the rest of this problem assume p = 2/3. Compute PNA(7).
(c) Assume out of 11 seats, 7 representatives will continue their terms, and only 4 are up for election. Among those 7 members, 4 of them belong to party B. What is the probability that party A has the majority of the Congress after the election?
Problem 5
The soccer team that I support plays 10 matches during the season. In each game, the winner gets 3 points, and the loser gets 0 points. In the case of a draw, each team gets a 1 point. The probability that my team wins in each game is p = 0.5, and they lose in each game with a probability q = 0.3. Therefore, the probability of a draw in each game is r = 0.2.
(a) What is the expected value of the number of points my team may get in each match?
(b) Let W be the total number of wins during the season. Write PW(w), the probability mass function of W.
(c) What is the probability that my team never loses during the season?
(d) What is the probability that the results of the first 6 games of the season are WWDLWL? (W=win, D=draw, L=lose)
(e) What is the probability that my team wins 3 games and loses twice during the first 6 games of the season?
(f) I usually go to the stadium to support my team, but I decided to stop going after my team lost 3 games. Let X
be the number of times I go to the stadium. What is the range of X? (g) Compute PX (10).
(h) Find the probability mass function of X.
(i) If my team has only 1 draw during the season, what is the probability that they get 22 points during the season?
Hint: How many games should my team win?
(j) What is the probability that my team receives exactly 20 points during the season?
Hint: You can partition the probability space based on the number of wins during the season.
(k) If my team has 20 points by the end of the season, what is the probability that they won exactly 5 games?