MA2510 - 2223B

MA2510 -2223B Solution

1. (15 marks) Five candidates, A, B, C, D and E have been scheduled for job interviews
at the same time. The hiring manager has scheduled the five for interview rooms 1,2, 3, 4 and 5 respectively. However, the manager’s secretary does not know this, so assigns the candidates to the five rooms in a completely random fashion. What is the
probability that
(a) All five candidates end up in the correct interview room?
(b) None of the five ends up in the correct room?
(c) None of A, B, C or D ends up in the correct room, given that E ends up in room
5?
2. (10 marks) A box contains 1 red ball, 2 black balls and 3 white balls. Consider the
experiment of drawing a ball from the box with replacement (after the draw, the ball
needs to be placed back into the bag again). We repeat such an experiment two-times.
LetX,Y and Z denote the number of balls drawn from the bag in red, black and white
respectively.
(a) Find the probability P{X = 1|Z = 0};
(b) Find the joint Probability Mass Function (PMF) of two random variables (X,Y).

3. (15 marks) The Probability density function of (X,Y) is

f (x, y) = C e^{- 2 x^{2} + 2 x y - y^{2}}, x, y \in (- \infty, \infty) .

(a) Find the constant C.
(b) Are X and Y independent?
(c) Find the conditional probability density function of f_{y|x}(y|z).
4. (10 marks)

\begin{aligned} Let X_{1}, X_{2}, \dots, X_{n} (n > 1) be a sequence of independent and identical distributed \\ random variables each having variance σ^{2} . Let Y = \frac{1}{n} \sum_{i = 1}^{n} X_{i}, \\ (a) Find V a r (X_{1} + Y) . \\ (b) Find C o v (X_{1}, Y) . \end{aligned}

(15 marks) Let X_1, X_2,... , X_n denote a random sample from the exponential distribution with unknown parameter A
(a) Find the estimate of \ by using method of moment estimation(MME) and maximum likelihood estimation(MLE).
(b) Are the estimates in (a) unbiased or not? Justify your answer.

6. (20 marks) A statics reports a sample of time (min) to serve a client in the Bank. Bank
believes that the data follows a normal distribution with known standard deviation
7. The sample size is 9, and the sample mean is 250.
(a) Is there compelling evidence for concluding that true average service time exceeds
200 min? Conduct a hypothesis test using a significance level 0.05 by calculating
the rejection region. .
(b) State the p-value for the test above.
(c) Suppose the true mean is 320, what is the type II error probability of the test in (a); that is, 8(320).
For (b) and (c), you can state your answers in terms of ®, the cumulative distribution
function of the standard normal distribution.
8. (15 marks) An innovative technology is used to measure the height that is known
to result in an error of measurement which is normally distributed with a standard
deviation of 0.08. suppose that the results of 8 independent measurements are
11.2, 12.4, 10.8, 11.6, 12.5, 10.1, 11.0, 12.2
(a) Give a 95% confidence interval of the true mean measurement level of height.
(b) Give a 95% Lower confidence interval.
(c) Give a 95% Upper confidence interval.
For (a), (b) and (c) the final answers can be stated using the notation z,or ze to
denote the critical value of the normal-distribution with n degrees of freedom and tail
probability a.
—-BND

Formula sheet
(1) Gaussian Integral: $$\int_{-\infty}^{\infty}\mathrm{e}^{-ax^{2}}\mathrm{d}x=\sqrt{\frac{\pi}{a}}.$$
(2) PDF of exponential random variable:

f (x) = {\begin{array}{rcl} λ e^{- λ x} & x > 0. \\ 0 & otherwise. \end{array}

Exponential r.v.:X ~ exp (Λ) ; Expectation: 1/Λ ; Variance: 1/(Λ^2)
(3) $$z_{0.05}=1.64,\quad z_{0.025}=1.96\\Phi(1)=0.8413$$