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UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

Department of Electrical and Computer Engineering

ECE302: Probability and Applications Final Examination, April 27, 2017, 9:30 am

Examiner: Konstantinos N. Platanioti s

Instructions:

You have two and one half hours (21 h) to complete the examination. There are SEVEN (7) problems in this paper. Answer ALL questions. The value of each question is marked beside the question. This test is a TYPE C examination. Calculator Type: 2.

STUDENT NAME: __________________ STUDENT NUMBER:

Problem Out of[ Score ]

1 8

2 5

3 5

4 7

5 8

6 7

7 10

Total 50

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ECE3021-11 S Probability and Applications: Final Examination Page 2 of25

Q.1. 8 marks In Canada, it is known that the probability of a human birth resulting in twins is about 0.012. Given that a human birth results in twins, the probability is 1 that they are identical ('mono zygomatic') twins, and the probability is that they are fraternal ('non mono zygomatic') twins. Identical twins are necessarily of the same sex, with male and female pairs of identical twins being equally likely to occur. Also, given that a pair of fraternal twins is born, the probability is that they are both females, the probability is that they are both males, and the probability is that there is one male and one female. Now consider the following events:

Event T: 'a Canadian birth results in twins'

Event 1: 'a Canadian birth results in identical twins'

Event F: 'a Canadian birth results in fraternal twins'

Event M: 'a Canadian birth results in twin males'

Find the numerical values of the following probabilities:

1(a) 1 mark P(I).

Provide your final answer here:

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1(b) 1mark P(F)

Provide your final answer here:

1(c) 1 mark P(M).

Provide your final answer here:

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1(d) 1mark P(F!t 1c).

Provide your final answer here:

1(e) 2marks P((Ifl!ti)T).

Provide your final answer here:

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1(1) 2marks P((IUM)JT)

Provide your final answer here:

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Q.2. 5 masks Given two random variables (r.v), X and Y assume that their covariance is given by

COV(X. Y).

2(a) 2 marks Show that VAR(X + Y) = VAR(X) + VAR(Y) + 2C0V(X, Y)

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2(b) 3 marks A fair die has two green faces, two red faces and two blue faces and the die is thrown once. Find the numerical value for COV(X, Y) if:

J1 if a green face is uppermost

0 otherwise

I1 if a blue face is uppermost

0 otherwise

Provide your final answer here:

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Q.3. 5 marks Suppose that X and Y are independent, Poisson random variables, with parameters )' and ,u respectively.

3(a) 2 marks Show that (X + Y) is Poisson with parameter (A +

Provide your final answer here:

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3(b) 2 marks Are X and (X + Y) independent. Justify your answer.

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3(c) 1 mark What is the conditional probability P(X = r(X + Y) = n) where r = 0, 1. n.

Provide your final answer here:

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Q.4. 7 marks Let X have cumulative distribution function (CDF):

0 ifx<0

F(x) = 0.5x if 0<r<2

1 ifx>2

and let Y = X2. Find

4(a)lmark P(O.5<X<1.5).

Provide your final answer here:

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4(b)1mark P(1<X<2).

Provide your final answer here:

4(c) 1 mark P(Y < X). for Y =X2

Provide your final answer here:

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4(d)1mark P(X<2Y)

Provide your final answer here:

4(e) 1rnark P((X+Y)<).

Provide your final answer here:

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4(f) 2 marks The cumulative distribution function of r.v. Z = /ik if 0 < < (2)2'.

Provide your final answer here:

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Q.5. 8 marks Let X be a discrete random variable with

X

P(X) I 0.3 0.5 0.2

2 marks Find the mean mx, variance 4, and standard deviation tTx of the random variable X.

Provide your final answer here:

2 marks Find the distribution (CDF), mean my. variance 4, and standard deviation ay of the random variable Y = X3 .

Provide your final answer here:

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5(c) 2 marks Find the distribution (CDF), mean my, variance 4, and standard deviation cry of the

random variable Y = 2'.

Provide your final answer here:

5(d) 2 marks Find the distribution (CDF), mean my, variance 4, and standard deviation cry of the

random variable Y = X 2 + 3X +4.

Provide your final answer here:

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Q.6. 7 marks Let X and Y have the joint probability density function (pdf) fxy(x. y) = cx(y -

0 < X <y< 00.

6(a) 2 mark Determine c.

Provide your final answer here:

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6(b) 2 marks Show that fxiy(xy) = 6x(y - x)y 3, 0 x ~ y.

Provide your final answer here:

6(c) 2 marks Show that fyx(yx) = (y - x)ex_Y, 0 <x <y < 0—c-

Provide your final answer here:

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6(d) 0.5 mark Show that E(XY = y) = 4y.

Provide your final answer here:

6(e) 0.5 mark Show that E(YX = .r) = .r + 2.

Provide your final answer here:

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Q.7. 10 marks To assess whether there is a genetic predisposition to becoming a cigarette smoker, epi-demiologic studies have been done in which sets of identical ('mono zygomatic') twins separated at birth and raised to adulthood in totally different environments are located and personally interviewed regarding their current smoking habits. For the jth adult member of such a set of identical twins

(i = 1,2), let Xi = 1 if that adult member is currently a smoker, and let Xi = 0 if that adult member

is currently not a smoker. The UofT bio-statisticians do not consider X1 and X2 independent ran-dom variables. They suggest using the following discrete 'distribution' px1,x2 (XI , x2) for X1 and

X9, where O>0. and 0<ir< 1.

P(X1 = X2 = 0) = K(1 - 7)2

P(X1 = X 2 = 1) = K7V 9

P((X = 1) fl(X2 = 0)) = = 0) fl(X2 = 1)) = K7r(1 - 7r)0

7(a) 2 marks Find the value of K which makes px1,x2 (xi, X2) a valid discrete distribution. Justify

your answer.

Provide your final answer here:

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7(b) 1 mark Find the marginal distribution for Xi.

Provide your final answer here:

7(c) 1 mark Find the marginal distribution for X2.

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7(d) 1 mark Find the explicit expression for P(X1 = 1X2 = 1).

Provide your final answer here:

7(e) 2 marks Find the explicit expression for E(X1 IX2 = 1) and VAR(XX2 = 1).

Provide your final answer here:

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7(f) 1 mark Find the explicit expressions for COV(X1, X2).

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7(g) 2 marks Find the explicit expression for correlation Q(X1, X2). For what specific set of values is this correlation, positive, negative or zero ?

Provide your final answer here:

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(Extra worksheet)