STAT2550 Spring 2023 Practice Midterm
June 13, 2023
Question 1. Suppose that a fair coin is tossed three times.
(a) What is the probability of two or more heads given that there was at least one head?
(5 marks)
(b) What is the probability of two or more heads given that there was at least one tail? (5
marks)
Question 2. Suppose two dice are rolled and the face values sum to six. What is the
probability that at least one of the dice is a three? (5 marks)
Question 3. Let X ∼ Exponential(λ). Compute the density fY (y) of Y = X 3 . (5 marks)
Question 4. Consider two random variables X and Y with joint probability function

1/9, x = −2, y = −4





2/9, x = −2, y = 5



3/9, x = −2, y = 9
p(x, y) =

1/9, x = 0, y = 9





2/9, x = 4, y = 9



0, otherwise.
(a) Compute P (X = 0|Y = 9) (2 marks)
(b) Compute P (Y = 5|X = −2) (2 marks)
(c) Are X and Y independent? (2 marks)
Question 5. Consider two random variables X and Y with joint probability function
1

1/7, x = 5, y = 0





1/7, x = 5, y = 1



1/7, x = 5, y = 2
p(x, y) =

3/7, x = 8, y = 5





1/7, x = 8, y = 6



0, otherwise.
(a) Compute E(X) and V ar(X). (2 marks)
(b) Compute Cov(X, Y ). What does its value tell you about the relationship between X
and Y ? (2 marks)
(c) Compute V ar(X) and V ar(Y ). (2 marks)
Question 6. Phone calls are received at an office as a Poisson process with parameter λ = 2
per hour. If there is no-one in the office for a period of 10 minutes, what is the probability
that the phone rings during that time? (3 marks)
2