Final Exam April Winter 2019, questions and answers PDF

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McGill University April 2019Faculty of Science Final examinationProbabilityMATHThursday, April 18th, 2019 Time: 9 am - 12 pmExaminer: Prof. D. Wolfson Associate Examiner: Prof. M. AsgharianINSTRUCTIONS Answer all questions on the question book itself. It is important to show your reasoning. The mark...

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McGill University Faculty of Science

April 2019 Final examination

Probability MATH323 Thursday, April 18th, 2019 Time: 9 am - 12 pm

Examiner: Prof. D.B. Wolfson

Associate Examiner: Prof. M. Asgharian

INSTRUCTIONS 1. Answer all questions on the question book itself. 2. It is important to show your reasoning. 3. The marks allocated are shown in [brackets] 4. Language dictionaries are permitted. 5. This is a closed book exam. 6. Hand calculators are permitted. 7. Crib sheets are not permitted. This exam comprises 15 pages including the cover page. Please PRINT your name and provide your student number:

LAST NAME:

FIRST NAME:

STUDENT NUMBER:

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Total

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Question 1 [10 marks] i) [5 marks] Let A and B be two arbitrary events. Prove that P(A ∪ B) = P(A) + P(B) − P(A ∩ B). [A Venn diagram is not sufficient] Pn S ii) [2 marks] Let A1 , A2 , ..., A n be n arbitrary events. Assuming that P( ni=1 Ai ) ≤ i=1 P(Ai ), prove

that:

n+1 [

P(

i=1

Ai ) ≤

n+1 X

P(Ai ).

i=1

iii) [3 marks] Suppose that 40 percent of ticks carry the organism that causes Lyme disease, and 10 percent carry the Zika virus. Further of those ticks that carry the Zika virus, 50 percent carry Lyme disease. What is the probability that a randomly selected tick carries at least one of these two organisms?

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Question 2 [10 marks] A drug smuggler inserts 5 bottles of Fentanyl, misslabeled as penicillin, into every box of 20 identical bottles. That is, each box of 20 bottles includes bottles of genuine penicillin mixed with 5 bottles of Fentanyl. A customs officer randomly picks 10 boxes for inspection from a large consignment. She randomly selects 7 bottles from each of these boxes and sends them for chemical analysis. What is the probability that exactly 4 of the boxes yield at least 3 bottles of Fentanyl among the 7 sent for chemical analysis? Leave your answer in unsimplified form. [Hint: Do this problem in two main steps]

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Question 3 [10 marks] A chest has 3 drawers. In each drawer there are 2 boxes. In one of the drawers one of the boxes contains a silver coin and the other box contains a gold coin. In one of the drawers each box contains a silver coin, and in the last drawer each box contains a gold coin. A drawer is selected at random and then one of the boxes in that drawer is randomly selected and opened. It is found to contain a silver coin. Using conditional probability and The Law of Total Probability, find the probability that the remaining box contains a gold coin.

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Question 4 [20 marks] Suppose that the number of telephone calls N , that arrive at a call centre in the time interval [0, s] has a Poisson distribution with parameter λs. That is, PN (k) = P(N = k ) =

(λs)k e−λs , for k = 0, 1, 2, ... k!

i) [5 marks] Using the definition of expected value, prove that E(N ) = λs. ii) [5 marks] Prove that the moment generating function of N is MN (t) = exp (λs(et − 1)) for −∞ < t < ∞. iii) [5 marks] Suppose there are two call centres that receive calls independently. The number of calls in [0, s] for the two call centres are Poisson random variables with expected values λ1 s and λ2 s. Using (ii), find an expression for the probability that the total number of calls in [0, s] for the call centres combined will be at least 2.

iv) [5 marks] Consider just the 1st call centre. Let X be the time at which the 1st call arrives. Show that the cumulative distribution function of X is given by 8 >

:0, for x < 0

[Consider the complement of the event {X ≤ x}c = {X > x}. If Nx = {no. of calls in [0, x]} how are Nx and {X > x} related?]

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Question 5 [10 marks] A computer program consists of 100 steps. The time to run step i, for i = 1, 2, ..., n, is a random variable Xi , whose mean is 0.01 sec and standard deviation is 0.001 sec. Suppose that the times to complete each of the steps are independent and identically distributed. Find the approximate probability that the average running time for this 100-step program is at least 0.0102 sec.

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Question 6 [10 marks] i) [5 marks] Let Z ∼ N (0, 1). That is ✓ ◆ 1 1 fZ (z) = √ exp − z 2 for − ∞ < z < ∞. 2 2π Prove that Y = Z 2 ∼ χ12 . That is fY (y) =

8 > < √1

π

y √1 e− 2 , 2y

> : 0,

for y ≥ 0 for y < 0

ii) [5 marks] Suppose that the distance in meters between flaws in a roll of material is a random variable with an Exponential distribution with parameter 10. That is, its cumulative distribution function is given by FX (x) =

8 x >

:0,

for x < 0

If a quality control inspector finds no flaw in a 9 meter length from the previous flaw, what is the probability that the next flaw will occur between 9 and 12 meters?

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Question 7 [13 marks] Suppose that a random variable Y has a probability density function given by 8 >

:0, elsewhere

i) [2 marks] Find c.

ii) [4 marks] Find the cumulative distribution function FY (y). iii) [3 marks] Find Var(Y ). ⇤ ⇥ iv) [4 marks] Let d be a constant. Show that E (Y − d )2 is minimized when d = E(Y ). Hence, write down ⇤ ⇥ the minimum value of E (Y − d )2 .

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Question 8 [13 marks] Suppose that the joint probability density function (pdf ) of Y1 and Y2 is given by, 8 >

: 0, elsewhere

i) [3 marks] Find the probability density function of Y2 , fY2 (y2 ). By recognizing this pdf, write down E(Y2 ). ii) [3 marks] Find P(− 12 ≤ Y1 ≤ 21 , 0 < Y2 ≤ 14 ) iii) [4 marks] Assume that the pdf of Y1 is given by 8 >

:0, elsewhere

Find Cov(Y1 , Y 2 ).

iv) [3 marks] For any two continuous random variables X and Y whose expectations exist, prove that E(X + Y ) = E(X) + E(Y ).

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