ECE2191

Probability Models in Engineering

Mock Exam

Information

This is an open-book exam. You are permitted to use lecture notes and calculators to complete the questions.

This exam consists of 7 questions.

Please answer all questions.

You have 2 hours and 10 minutes to complete the exam.

The exam has the following sections:

Section A: Essay and Multipart Questions

Answer the question in section A by entering text online in the dedicated space below each

question.  Note that these questions may have multiple parts and all parts should be attempted.

When writing in text boxes:

please write fractions as a/b

please write exponents as a^(b-c) for ab-c

please use any of the following for complement of A, Ac: A', Ac or A^c

You can use any of the following notations for combination: n choose r ,   C(n,r)  or   n C r

You can use any of the following notations for Px or Px,y, etc: Px, P_x, Pxy, P_xy, P_{x,y}, You can use any of the following notations for or fx,y, etc: f_x, f_xy, f_{x,y},

For ≤ and ≥ use <= and >= or insert these characters: ≤ and ≥

For ± write: +/- or insert this character ±

For square root of a, write sqrt(a) or sqrt{a}, or a^{1/2} or a^(1/2)

For Phi Φ function write Phi(x) or insert this character: Φ

For sigma σ write sigma or insert this character: σ

For summation of i from i=1 to n, write sum_{i=1}^{n} (i) or sum{i=1 to n} (i) or insert character Σ_{i=1}^{n} (i) or write sum of i from i=1 to n.

For approximate results write ~ or insert character ≈

. Section B: Hand-written/drawn response Questions

Answer all of the hand-written/drawn response questions in section B on your own pieces of paper. All questions in section B should be attempted. Please clearly label each blank piece of paper with your Student ID and the question number (and subpart of the question, if applicable). Please do not write your name on the paper. You will have time at the end of your exam to upload photographs of your answer sheets.

If you believe there is an error or a question is unclear, clearly state any assumptions you need to make and proceed.

Section A: Essay and Multi part Questions

Question 1

My partner and I are one of 10 married couples at a dinner party. The 20 people are given random seats around a round table. What is the probability that I am seated next to my spouse?

Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

Question 2

The PDF of a continuous random variable X is shown below:

The random variable Y is related to X as follows:

Answer the following questions. Explain your results in full by including all reasoning, formulae used  and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

2a) What is the possible value of a?

2b) What is the probability of event X=1?

2c) Find the probability mass function of Y.

Section B: Hand-written/drawn response Questions

Question 3

In a binary transmission system, the transmitted signalX is either -1 or 1, depending on whether sending a “0” bit or a “ 1” bit, respectively. The received signal Y is corrupted by an additive noise N with a zero-mean Gaussian distribution with variance σ2  = 1/16, i.e. Y = X + N. Assume that “0” bits are two times as likely as “ 1” bits.

a) Assume that the receiver decides a “0” was transmitted if the observed value of y satisfies

fY (y|X = −1)P(X = −1) > fY (y|X = 1)P(X = 1) and it decides a “ 1” was transmitted otherwise. Find a threshold T where this decision rule becomes equivalent to deciding “0” if y < T and deciding “ 1” if y ≥ T.

b) What is the overall probability of error?

Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

You may use the following Q-function table:

Question 4

Anytime Fae throws a Frisbee, her dog catches the Frisbee with probabilityp = 0.1, independent of whether the Frisbee is caught on any previous throw. When the dog catches the Frisbee, it runs away with the Frisbee, never to be seen again. Fae continues to throw the Frisbee until the dog catches it.

Let X denote the number of times the Frisbee is thrown.

a) Find an expression for PMF of X?

b) What is the probability that Fae will throw the Frisbee more than three times?

c) Find the expected value of X.

d) Now assume that her dog always returns the Frisbee after catching it, so that Fae can throw it again. Every time her dog returns with the Frisbee, Fae rewards her with a meatball. Find an expression for PMF of the number of Meatballs given to the dog if Fae throws the Frisbee 100 times.

Question 5

Answer both of the following:

(a) Let Z = aX + bY, where a and b are constants, and X and Y are continuous random variables. Show that E(Z) = E(aX + bY) = aE(X) + bE(Y).

(b) Let G = aF + b, where a and b are constants, and F is a continuous random variable. Find the covariance of F and G.

Justify your answers by including all reasoning and/or formulae used to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

Question 6

Consider an experiment of drawing randomly three balls from a bowl containing two red, three white and four blue balls. Let (X,Y) be a pair of random variables, where X and Y denote, respectively, the number of red and white balls chosen.

Answer all of the following parts to the question:

(a) Find the joint probability mass function of (X,Y).

(b) Find the marginal probability mass functions of X and Y.

(c) Are X and Y independent?

Justify your answers by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

Question 7

The diameter of a particular brand of cricket ball is approximately normally distributed, with a mean of 7.12cm and a standard deviation of 0.075cm. If you select a random sample of nine cricket balls:

(a) What is the probability that the sample mean is less than or equal to 7.07cm?

(b) What is the probability that the sample mean is between 7.095cm and 7.145cm?

(c) The probability is 60% that the sample mean will be between what two values symmetrically distributed around the population mean?

Answer all of the above parts to the question. Justify your answers by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)