ECE2191 Probability Models in Engineering
Tutorial 9: Multiple RVs
Second Semester 2021
1. Let X, Y and Z be three jointly continuous random variables with joint PDF
Find the joint PDF of X and Y; i.e., fX,Y(x, y).
2. The random variables X and Y are described by a joint PDF which is constant within the unit area quadrilateral with vertices (0,0), (0,1), (1,2), and (1,1).
(a) Are X and Y independent?
(b) Find the marginal PDFs of X and Y.
(c) Find the expected value of X + Y.
(d) Find the variance of X + Y.
3. Let X = [X1, X2, ..., XN ]T be a random vector with joint pdf fX1,X2,...,XN (x1, x2, ..., xN). Find the joint pdf of Y = AX + b, where A is an N × N diagonal matrix given by
4. Let X and Y have the joint PDF
fX,Y(x, y) = xe-x(1+y) for x > 0, y > 0.
Find the marginal PDF of X and of Y.
5. The joint density function of X and Y is given by
f (x, y) = xe-x(y+1) x > 0, y > 0.
(a) Find the conditional density of X, given Y = y; and that of Y, given X = x.
(b) Find the density function of Z = XY.
6. Each morning, hungry Harry eats some eggs. On any given morning, the number of eggs he eats is equally likely to be 1, 2, 3, 4, 5, or 6, independently of what he has done in the past. Let X be the number of eggs that Harry eats in 10 days. Find the mean and variance of X .
7. There are n components. On a rainy day, component i will function with probability pi; on a non-rainy day, component i will function with probability qi, for i = 1, ..., n. It will rain tomorrow with probability α . Calculate the conditional expected number of components that function tomorrow, given that it rains.
8. Let X1 , X2, and X3 be three Geometric random variables with parameter p.
(a) Derive the joint PMF, joint CDF, and the covariance matrix of X1, X2, and X3 assuming that they are independent.
(b) Derive the joint PMF, joint CDF, and the covariance matrix of X1, X2, and X3 assuming that X1 = X2 and X3 is independent of X1 .
9. The random variables X1, X2, and X3 have the following joint Gaussian distribution
(a) Find the marginal pdf of X1 and X2; i.e., fX1,X2 (x1, x2).
(b) Find the marginal pdf of X1 and X3; i.e., fX1,X3 (x1, x3).
(c) Find the conditional pdf, mean, and variance of X2 given X1 and X3 .
10. Let be a normal random vector with the following mean vector and covariance matrix:
Let also
(a) Find P(0 ≤ X2 ≤ 1).
(b) Find the expected value vector of Y, mY = E[Y].
(c) Find the covariance matrix of Y, CY.
(d) Find P(Y3 ≤ 4).
11. [Optional] Suppose that the expected number of accidents per week at an industrial plant is four. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2. Assume also that the number of workers injured in each accident is independent of the number of accidents that occur. What is the expected number of injuries during a week?
12. [Optional] Let X and Y be two independent uniform random variables between 0 and 1. Let the random vectors U and V be defined as
Determine whether CUU and CVV are positive definite.