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MAT 3172

Practice questions

2022

1. a) (5 marks) State the deﬁnition of the conditional density function of Y given that X = x, in the case of jointly continuous random variables X and Y .

b) (10 marks) Let X and Y be jointly continuous random variables with joint density given by:

f (x, y) = , (y2 ≠ ≠y 〕x 〕y

Find the conditional density of Y given that X = x.

2. a) (5 marks) a) State the deﬁnition of independent random variables X and Y .

b) (10 marks) Let X and Y be jointly discrete with joint probability mass function given by the following table:

x\y 0 1

0.2

0.4

Give the joint probability mass function of the variables U = X + Y and V = XY . Are U and V independent? Justify your answer.

3. A class contains M female students and N male students. A group of n students is selected, for some positive integer n 〕M + N . Let X be the number of female students in the group.

a) (5 marks) Calculate P (X = k). What are the possible values k of X?

b) (5 marks) Calculate E(X).

c) (5 marks) Calculate E(X2 ).

Hint: Let Ai be the event that the i-th selected student is a female, and Ii be the indicator function of the event Ai , for i = 1, . . . , n. Express X and X2 using the indicator functions Ii .

4. a) (5 marks) State the deﬁnition of the covariance between two random variables X and Y .

b) (5 marks) Let X and Y be jointly continuous with joint density

f (x, y) = exp , ≠ } , ≠之 < x < 之, ≠之 < y < 之,

for some a > 0. Let U = aX + bY and V = aX ≠ bY . Find the covariance between U and V .

Hint: Use the fact that the density of a normal random variable X with mean µ and variance σ 2 is given by:

f (x) = exp , ≠ (xµ2)2 } , ≠之 < x < 之.

c) (5 marks) Let X← , . . . , Xn be independent random variables with a normal distribution with mean 0 and variance 1. Let

n n

U =z aiXi and V =z biXi ,

i』← i』←

for some real numbers ai , bi . Find the covariance between U and V .

5. a) (5 marks) Give the deﬁnition of the random variable E[X|Y].

b) (5 marks) Let X and Y be independent Poisson random variables with mean λ, respectively

µ i.e.

P (X = k) = e −入 , P (Y = k) = e −人 k = 0, 1, 2, . . .

Find E[X|X + Y]. Hint: Calculate ﬁrst E[X|X + Y = n].

c) (5 marks) For coming to the university, a student has 3 choices: by car, by bus, or by bike. The average time for getting to the campus by car is 15 minutes, by bus is 20 minutes, and by bike is 25 minutes. Depending on the weather and his schedule, the student will take the car, the bus, or the bike with probabilities 0.05, 0.10, respectively 0.85. What is the average time for getting to the campus, regardless of the means of transportation?

6. a) (5 marks) Give the deﬁnition of the moment generating function of a random variable X .

b) (5 marks) Let X be a negative binomial random variable with parameters (r, p), where r 女 1 is an integer and 0 〕p 〕1, i.e.

P (X = k) = ╱ r(k) 1(1) \ pr (1 ≠ p)k −r , for k = r, r + 1, r + 2, . . .

Find the moment generating function of X .

c) (5 marks) Let X and Y be independent random variables with negative binomial distribu- tions with parameters (r← , p), respectively (r2 , p). Find the distribution of X + Y .

Hint: Use b).

Formula Page

Theorem. Let X← and X2 be jointly continuous random variables with joint density fk1 /k2 (x← , x2 ). Let g ← : R2 女 R and g2 : R2 女 R be two functions with the following properties:

1. The system of equations:

g ← (x← , x2 ) = y ←

g2 (x← , x2 ) = y2

has a unique solution x ← = h← (y← , y2 ) and x2 = h2 (y← , y2 ).

2. The functions g ← and g2 have partial derivatives, such that:

J(x← , x2 ) = │ │ 0, for all x ← , x2 .

Then, the random variables Y← and Y2 , deﬁned by:

Y← = g← (X← , X2 ) and Y2 = g2 (X← , X2 ),

have joint density:

Corollary. Let X← and X2 be jointly continuous random variables with joint density fk1 /k2 (x← , x2 ). Then, the random variables Y← and Y2 , deﬁned by: