MATH20701 2021
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MATH20701
1.
Let X,Y be a bivariate random variable with joint probability density function given by
fX,Y (x,y) = 
where A > 0 is a constant.
(i) Show that A = 4. [4 marks]
(ii) Find the marginal probability density function of X . [4 marks]
(iii) Find the marginal probability density function of Y . [4 marks]
(iv) Find P(X < 2Y | X < 2). [8 marks]
[End of Question 1; 20 marks total]
2. Let X and Y be discrete random variables, which are independent of each other, with probability mass functions given by
P(X = k) = (
Let Z = min(X,Y).
(i) Prove that c = 
.
P(Y = k) = 
k = 2, 3, . . .
otherwise, [5 marks]
(ii) For k 2 {1, 2, . . .} find P(X > k) and P(Y > k). [5 marks]
(iii) For k 2 {1, 2, . . .} find P(Z > k). [5 marks]
(iv) Hence, or otherwise, find the probability mass function of Z . [5 marks]
[End of Question 2; 20 marks total]
3. Let X and Y be continuous random variables, which are independent of each other, with probability density functions given by
fX (x) = 
fY (y) = 
(i) Find the probability density function of X + Y . [6 marks]
(ii) Let Z = X/Y . Find the joint probability density function of (Z,Y). [8 marks]
(iii) Hence, or otherwise, find the probability density of Z . [6 marks]
[End of Question 3; 20 marks total]
4. The joint probability density function of random variables X and Y is f(x,y) = 
(i) Derive the conditional probability density functions f(x | y) and f(y | x), stating clearly for which values of y and x they are respectively defined. [7 marks]
(ii) Determine E[Y | X = 1]. [7 marks]
(iii) Calculate Cov(X,Y). [6 marks]
[End of Question 4; 20 marks total]
5. a) Let U denote a random variable with probability density function
f(u) = { 
(i) Define the moment generating function of a continuous random variable. [3 marks]
(ii) Derive the moment generating function MU (t) of the random variable U. [4 marks]
(iii) Write down the power series expansion of MU (t) and hence find the variance and third moment of U. [3 marks]
b) The random variable X has probability density function
f(x) = { 
where ✓ > 0 is a constant.
(i) Show that, for t < ✓ , the moment generating function of X is
✓
MX (t) = [5 marks]
(ii) The number Z of tasks which a worker is required to complete has probability mass function
P(Z = k) = {0(q)k−1p
k = 1, 2, ...
elsewhere,
where p,q 2 (0, 1) are such that p + q = 1.
The tasks are of independent durations, each having the same distribution as X above. Z is independent of the durations of the tasks. Using generating functions or otherwise, determine to which family the distribution of the time taken to complete all the tasks belongs. [5 marks]
[End of Question 5; 20 marks total]
6. a)
(i) State Chebyshev’s inequality. [3 marks]
(ii) A telephone exchange handles, on average, 10,000 calls an hour with a variance of 2,000. Use Chebyshev’s inequality to derive a lower bound for the probability that the exchange will handle between 9,920 and 10,080 calls during a 1 hour period. [3 marks]
(iii) Assume now, that the exchange services a town of 12,500 people who each, independently, call the exchange with probability 0.8 during a given 1 hour period. Calculate the probability of the same event as in (ii) using the Normal approximation to the Binomial distribution. [4 marks]
b) Let Z1 and Z2 be independent random variables, each normally distributed with mean 0 and variance
1. Let Z = (Z1 ,Z2 )T and define
X = (X1 ,X2 )T = AZ
where
(i) Determine the covariance matrix of A = ✓ 
2 ◆ . [4 marks]
(ii) Stating any results to which you appeal, prove that the joint probability density function of X1 and X2 is
f(x1 ,x2 ) = 
exp{−(x1(2) + x2(2) + ^2x1 x2 )} − 1 < x1 ,x2 < 1. [6 marks]
[End of Question 6; 20 marks total]
2023-01-09