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ISyE 6739 Practice Test #2 Solutions

Summer 2010

1. Short Answer Questions.

(a) SupposeXhas p.d.f.f(x) = 5x4, 0 x 1. Find E[2X 5].

ANSWER: 2E[X] 5 = 10/3.

(b) IfXagain has p.d.f.f(x) = 5x4, 0 x 1, find Var(2X 5).

ANSWER: 4Var(X) = 0.0793.

(c) Suppose Xcan equal 1 or 2, each with probability 1/2. Find E[n(X)].

ANSWER: 12n(2) = 0.347.

(d) TRUE or FALSE? E[X2] (E[X])2.

ANSWER: True.

(e) TRUE or FALSE? IfCov(X, Y) = 0, then X and Yare independent.

ANSWER: False.

(f) Suppose that X and Y are independent Exponential() random variables.Find Var(XY) (yup the variance of the product).

ANSWER: E[X2Y2 ] (E[XY])2 = E[X2]E[Y2 ] (E[X])2(E[Y])2 = 3/4.

(g) What does i.i.d. mean?

ANSWER: independent and identically distributed.

(h) Bonus Question: If you were Dutch, what would Q-65 mean to you?

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ANSWER: They later became Golden Earring.

2. Suppose X and Y are two random variables such that E[X] = 3, Var(X) = 4,E[Y] = 2, Var(Y) = 9, and Cov(X, Y) = 2.

(a) Find the correlation between 2Xand 3Y.

ANSWER:

Corr(2X, 3Y) = Cov(2X, 3Y)Var(2X)Var(3Y)

= Cov(X, Y)Var(X)Var(Y)

=1/3.

(b) Find Var(X Y).

ANSWER: Var(X) + Var(Y) 2Cov(X, Y) = 17.

3. (10 points) Suppose X has p.d.f. f(x) = 3x2, 0 x 1. What is the p.d.f. ofY =

X?

ANSWER: The c.d.f. ofY is

G(y) = Pr(Y y) = Pr(X y)= Pr(X y2) =

y20

3x2 dx

= y6.

Thus, the required p.d.f. isg(y) = 6y5, 0 y 1.4. Suppose that f(x, y) =cxy if 0< y < x < 1.

(a) Find c.

ANSWER: Use 1 =10

x0 cxy dy dx to get c= 8.

(b) Find fX(x).

ANSWER:x

0 8xy dy = 4x3, 0< x

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(c) Find Corr(X, Y).

ANSWER: Using the usual methods, we have

E[X] = 4

5, Var(X) =

2

75, E[Y] =

8

15, Var(Y) =

11

225,

and

E[XY] = 10

x0

8x2y2 dy dx = 4

9.

Thus,

= E[XY] E[X]E[Y]

Var(X)Var(Y)= 0.492.

(d) Find f(y|x).

ANSWER: f(x, y)/fX(x) = 2y/x2, 0< y < x

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(b) IfX Pois(), find gX(s). (You may need the fact thatk=0yk/k! =ey.)

ANSWER:

gX(s) =k=0

skek

k! = e

k=0

(s)k

k! = e(s1).

(c) Suppose that X Pois(). Use the above two parts of the problem to findE[X].

ANSWER:

E[X] = d

dsgX(s)|s=1 = e(s1)

s=1

= .

6. (Short normal distribution questions Just write your answer.)

(a) Find z0.025.

ANSWER: 1.96.

(b) Find (1.85).

ANSWER: 0.9678.

(c) IfX N(2, 4), find Pr{X 0}?

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ANSWER: 2X1X2X3 Nor(2 1 1, 4 + 1 + 1 ) Nor(0, 6), so an easysymmetry argument says that the desired probability is 1/2.

7. Quickie Probability Questions Just Write Your Answer.

(a) IfXis a continuous random variable with p.d.f. f(x) = (1/2)ex/2 for x 0,find E[X].

ANSWER: 2.

(b) Suppose that Xhas m.g.f.MX(t) =pet +q. Whats E[X]?

ANSWER: MX(t) =pet|t=0=p.

(c) TRUE or FALSE? The Law of the Unconscious Statistician states that, ifg()is a continuous function andXis a random variable, then E[g(X)] =g(E[X]).

ANSWER: False.

(d) TRUE or FALSE? IfXand Y are independent continuous RVs, then

Pr(X Y) =

y

f(x, y) dxdy.

ANSWER: True.

(e) TRUE or FALSE? E[E(X|Y)] = E[X].

ANSWER: True.

(f) IfX

Exp(3), what is its moment generating function?

ANSWER: 3/(3 t), for t

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Further suppose that the conditional p.d.f. ofX given Y is

f(x|y) = 2x/y2, 0

x

y

1.

Show your work on the following questions.

(a) Find f(x, y).

ANSWER: f(x, y) =f(x|y)fY(y) = 8xy, 0 x y 1.

(b) Find fX(x).

ANSWER: fX(x) =1x f(x, y) dy= 4x(1

x2), 0

x

1.

(c) Use Part 8b to find E[X].

ANSWER: E[X] =10 xfX(x) dx= 8/15.

(d) Find E[X|y].

ANSWER: E[X|y] =y0 xf(x|y) dx= 2y/3, 0 y 1.

(e) Use Part 8d to find E[X]. (This should match your answer in Part 8b.)

ANSWER: E[X] = E[E[X|Y]] =10 E[X|y]fY(y) dy =10(8/3)y4 dy = 8/15.

(f) Find the correlation betweenXand Y.

ANSWER: After the usual algebra, we get E[X] = 8/15, E[X2] = 1/3,Var(X) = 11/225, E[Y] = 4/5, E[Y2 ] = 2/3, Var(Y) = 2/75.

Further, E[XY] =

10

1x 8x2y2 dy dx= 4/9, so that

= E[XY] E[X]E[Y]

Var(X)Var(Y)= 0.4924.

9. Given the following joint p.d.f.s (or other info), determine whether or not X andY mustbe independent. Just answer yes or no.

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(a) g(x, y) =2e(x+y),x >0, y >0.

ANSWER: Yes.

(b) f(x, y) =cxy, 0< y < x < 1.

ANSWER: No.

(c) f(x, y) =c(1 +x+y)2,x >0, y >0.

ANSWER: No.

(d) f(x|y) =fY(y) for all y.

ANSWER: No.

(e) Cov(X, Y) = 0.

ANSWER: No.

10. Suppose thatE

(X) = 3,E

(Y) = 2,Var

(X) = 5,Var

(Y) = 4, andCov

(X, Y) = 2.(a) Find E(2X+ 3Y).

ANSWER: 2E[X] + 3E[Y] = 12.

(b) Find Var(2X+ 3Y).

ANSWER: 4Var(X) + 9Var(Y) + 2Cov(2X, 3Y) = 4Var(X) + 9Var(Y) +12Cov(X, Y) = 32.

(c) Find the correlation betweenXand Y.

ANSWER: We have

= Cov(X, Y)Var(X)Var(Y)

=0.447.

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11. Short Answer Questions.

(a) SupposeXhas p.d.f.f(x) = 5x4

, 0 x 1. Find E[2X 5].ANSWER: 2E[X] 5 = 10/3.

(b) IfXagain has p.d.f.f(x) = 5x4, 0 x 1, find Var(2X 5).

ANSWER: 4Var(X) = 0.0793.

(c) Suppose Xcan equal 1 or 2, each with probability 1/2. Find E[n(X)].

ANSWER: 12n(2) = 0.347.

(d) SupposeXhas m.g.f.MX(t) = 0.3et + 0.7. Whats the distribution ofX?

ANSWER: Bern(0.3).

(e) IfXhas m.g.f. 4/(4 t), for t 5).

ANSWER: By the memoryless property, Pr(X > 7

|X > 5) = Pr(X > 2) =

e2/5.

12. Dont forget to practice questions from your favorite distributions, e.g., from Home-work 7 Bernoulli, binomial, geometric, negative binomial, Poisson, uniform, ex-ponential, etc., etc.