Question 1

Two students are enrolled in a psychology exam. Assume that the probability that student A passes the exam is 80%, and student B 60%. The event that student A passes the exam or not does affect the performance of student B.

(a) Compute the probability that at least one of the two students will pass the exam.

(b) If at least one of the two students passes the exam, what is the probability that student A passes the exam?

Question 2

In an NBA final, two teams A and B play a series of games, capped at 7. The first team who wins a total of four games wins the champion. Assuming the probability that team A wins against team B is 1/3, compute the probability that team A will win the champion.

Question 3

Suppose that 𝑋 is a random variable for which 𝐸(𝑋) = and 𝑉𝑎𝑟(𝑋) =  2 . Let 𝑐 be an arbitrary constant. Show that 𝐸[𝑋(𝑋 − 𝑐)].

Question 4

The cumulative probability function (CDF), 𝐹(𝑥), of a random variable 𝑋 is sketched below in Figure Q4. Find the corresponding probability.

(a) 𝑃(𝑋 = −1)

(b) 𝑃(𝑋 < 0)

(c) 𝑃(0 < 𝑋 < 3)

(d) 𝑃(1 < 𝑋 <2)

(e) 𝑃(𝑋 > 5)

Question 5

In the data set ‘RVX.csv’, there is a random variable 𝑋.

(a) Determine the following values of 𝑋 using R: mean, median, 25% quantile, 75% quantile, and variance.

(b) Plot the histogram of

Question 6

(a) Suppose that three random variables 𝑋1,𝑋2, 𝑋3 form a random sample from the continuous uniform distribution on the interval [0, 1]. Assume 𝑋1, 𝑋2,𝑋3 are
independent, calculate the expectation of 𝐸[(𝑋1 − 2𝑋2 + 𝑋3) 2 ].

(b) A particle is confined in a straight tunnel aligned in an east-west direction and it can move randomly by the step size of one or two units. For each movement, the probability is 𝑝 that the particle will move one unit to the west, the probability is 𝑞 that the particle will move two units to the east, and the probability is 1 − 𝑝 − 𝑞 that the particle will remain at the same place. (0 ≤ 𝑝 ≤ 1, 0 ≤ 𝑞 ≤ 1, 0 ≤ 1 − 𝑝 − 𝑞 ≤ 1). A movement is independent of another. Calculate the expectation of the position of the particle after 𝑛 movements, assuming that the position of starting point is 0.

Question 7

Suppose that a pair of fair dice is rolled 120 times. Let 𝑋 be the number of rolls on which the sum of the two numbers is 12. Show your working details. You may use R to verify your answer.

(a) Find the probability when 𝑋 = 3 approximately using Poisson approximation.

(b) Compute the actual probability from a binomial distribution. Comment on the accuracy of the Poisson approximation