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 not affect the performance of student B.

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

(5 marks)

(b)If at least one of the two students passes the exam, what is the probability that student A passes the exam? (5 marks)

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. (10 marks)

Question 3

Suppose that is a random variable for which () = and () = 2. Let be an arbitrary constant. Show that [( )] = ( ) + 2.

(10 marks)

Question 4

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

Figure Q4

(a)( = 1) (3 Marks)

(b)(0) (3 Marks)

(c)(0<3) (3 Marks)

(d)(1<2) (3 Marks)

(e)(>5) (3 Marks)

Question 5

In the data set 'RVX.csv' there is a random variable .

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

quantile, and variance. (10 Marks)

(b) Plot the histogram of . (5 Marks)

Question 6

(a) Suppose that three random variables 1,2,3 form a random sample from the continuous uniform distribution on the interval [0, 1]. Assume 1, 2, 3 are independent, calculate the expectation of [(1 22 + 3)2].

(10 marks)

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

(10 marks)

Question 7

Suppose that a pair of fair dice are rolled 120 times. Let 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 = 3 approximately using Poisson approximation.

(10 marks)

(b)Compute the actual probability from binomial distribution. Comment on the accuracy of Poisson approximation. (10 marks)