## Description

1. Suppose the random variable X has probability density function

fX(x) = 2

x2

, x > 2

0, x ≤ 2.

(1)

(a) Let Y = log(X − 1), where log denotes the natural logarithm. Find the

probability density function of Y . [1 mark]

(b) Determine the quantile function of X. [1 mark]

(c) Hence, write a small function in R or MATLAB to simulate random variables

from the probability density function (1). Supply your code. [1 mark]

2. Consider a system comprising three components. The system requires at least two

out of the three components to be working for the system to work. Components

fail independently and the time to failure for each component has an exponential

distribution with a mean of one year.

(a) Determine an expression for the probability that the system is working at time

t. Hence, give the probability density function for the time to failure of the

system. [1 mark]

(b) Determine the mean and variance of the time to failure for the system.

[2 marks]

(c) Determine the probability that component one in the system is still working

at time t given the system is working at time t. What is the limiting value as

t → ∞? [2 marks]

(d) Suppose now that the system has one spare component which can be used to

replace a component in the failed system. The time to failure of the spare component is also an exponential distribution with a mean of one year. Consider

the following repair regime. As soon as the system fails, the one of the failed

components in the system is replaced by the spare and the system continues

to function as long as there are at least two components are working. Simulate the processes in R or MATLAB. You may wish to use the function sort in

MATLAB or R. Provide a histogram of the failure times for the repaired system

and an estimate of the mean time to failure for the repaired system. Supply

your code. [2 marks]

1

3. A pair of random variables (X, Y ) has a joint probability distribution in which the

marginal distribution of X is Exp(1) and the conditional distribution of Y given

X = x is Exp(x).

(a) Determine the marginal probability density function of Y . [1 mark]

(b) Determine the conditional expectation E[X|Y = y]. [1 mark]

4. Suppose Z has a bivariate normal distribution with mean 0 and covariance matrix

equal to the identity matrix. Determine a vector a and matrix B such that Y =

a + BZ has a bivariate normal distribution with

µ =

3

−2

and Σ =

5 13

13 41

Note that there may be more than one solution. Hint: You may wish to set B12 = 0,

though this is not the only way to solve this problem. [3 marks]

Total [15 marks]

2