NA387exam2-07.SOLUTIONS Pmf& Cdf Question

7/30/2019 NA387exam2-07.SOLUTIONS Pmf& Cdf Question

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NA387(3) Exam #2 Room 1024 FXB March 30, 2007, 2-3:30 PM

SOLUTIONS

Problem 1 (15%): Fill-in the blanks questions: (one point per blank correctly filled,no penalty for wrong answers)

1. Any random variable whose only possible values are 0 and 1 is called a __________ random variable; a special name given after the individual who firststudied it.

ANSWER: Bernoulli

2. The tension (psi) at which a randomly selected tennis racket has been strung is anexample of a __________ random variable.

ANSWER: continuous

3. The probability mass function of a discrete random variable X is defined as p( x) =ax for x = 1,2,3,4, then the value of a is __________.

ANSWER: .10

4. The cumulative distribution function F ( x) of a discrete random variable X is: F (1)

= .4, F (2) = .7, F (3) =.9, and F (4) = 1, then the value of the probability massfunction p( x) at X = 3 is _________.

ANSWER: .20

4. The probability mass function p( x) of a discrete random variable X is p(0) = .15, p(1) = .30, p(2) = .20, p(3) = .10, and p(4) = .25, then the value of the cumulativedistribution function F ( x) at X = 2 is __________.

ANSWER: .65

6. If the expected value of a discrete random variable X is E ( X ) = 5, then E (2 X + 3)is __________.

ANSWER: 13

7. Let X be a discrete random variable with E ( X ) = 4.5 and 2( ) E X = 26.25, then thevariance of X is V ( X ) = __________.

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ANSWER: 68. If the assembly time for a product is uniformly distributed between 15 to 20

minutes, then the probability of assembling the product between 16 to 18 minutesis __________.

ANSWER: .49. The cumulative distribution function F ( x) for a continuous random variable X isthe area under the density curve to the __________ of x.

ANSWER: left

10. If Z is a standard normal random variable, then( 2.0) P Z = __________. ANSWER: .0228

11. Let the random variables1 2 ,....., n X X X have mean values1 2, ,....., ,n and definethe random variableY as 1 1 2 2 1 2, where , , ....,n n nY a X a X a X a a a= + + +L aren numericalconstants. Then, E (Y ) = __________.

ANSWER: 1 1 2 2 n na a a + + + +L .12. If are independent random variables with variances2 2 21 2, ,...., ,n and the random

variable 1 1 2 2 ..... ,n nY a X a X a X = + + + thenV (Y ) = __________.

ANSWER:2 2 2 2 2 21 1 2 2 n na a a + + +L

13. If 1 2and X X are independent random variables with variances2 21 2and , respectively, then

1 2(3 5 )V X X + = __________.

ANSWER: 2 21 29 25 +

14.15. Let X and Y be two continuous random variables, and let f ( x, y) be their joint probability density function. The marginal probability density function of Y isobtained by __________ f ( x, y) over all possible values of variable __________

ANSWER: integrating, X

Problem 2 (50%) Multiple Choice Questions, 2 points per correct answer, nopenalty for wrong answers. No explanations necessary.

1. Which of the following statements are not correct?

A. A discrete random variable X can assume only a finite number of possiblevalues.

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B. A discrete random variable X is any random variable whose possible valueseither constitute a finite set or else can be listed in an infinite sequence inwhich there is a first element, a second element, and son on.

C. A random variable X is said to be continuous if its set of possible valuesconsists of an entire interval on the number line.

D. Number of students in a statistics class next year is an example of a discreterandom variable.

ANSWER: A

2. Which of the following statements are not correct?

A. The study of continuous random variables requires the continuousmathematics of the calculus - integrals and derivatives.

B. To study basic properties of discrete random variables, only the tools of discrete mathematics -summation and differences - are required.

C. The number of movies you watched last year is an example of a continuousrandom variable.D. In general, each outcome of an experiment can be associated with a number

by specifying a rule of association.

ANSWER: C

3. Which of the following statements is not an example of a discrete random variable?

A. The number of female respondents to a questionnaireB. The age of female respondents to a questionnaireC. The number of sales a salesperson makes per year D. The number of school-age children a working woman has

ANSWER: B

4. Which of the following statements is not an example of a continuous randomvariable?

A. The weight gain in pounds per month for a calf B. The price for cheesecake in New York Style cheesecakeC. The time it takes you to finish this statistics testD. The number of typos on a randomly chosen page of a book

ANSWER: D

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5. The probability mass function of a discrete random variable X is defined as p( x) = x/10 for x = 0,1,2,3,4. Then, the value of the cumulative distribution function F ( x)at x= 3 is

A. .10

B. .30C. .60D. .90

ANSWER: C

6. The cumulative distribution function F ( x) of a discrete random variable X is given by F (0) = .30, F (1) = .70, F (2) = .90, and F (3) = 1.0, then the value of the probability mass function p( x) at x = 1 is

A. .30

B. .40C. .20D. .80

ANSWER: B

7. Let X be a discrete random variable withV ( X ) = 3.70, thenV (2 X ) is

A. 13.69B. 5.70C. 14.80D. 7.40

ANSWER: C

8. Let X be a discrete random variable withV ( X ) = 8.6, thenV (3 X + 5.6) is

A. 77.4B. 14.2C. 83.0D. 31.4

ANSWER: A9. Which of the following statements are true?

A. For any discrete random variable X and constantsa andb, E (aX +b) = (a + b). E ( X )

B. For any discrete random variable X and constantsa and b, V (aX+b ) =2( ) ( )a b V X + .

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C. If a constantc is added to each possible value of a discrete random variable X ,then the variance of X will be shifted by that same constant amount.

D. If a constantc is added to each possible value of a discrete random variable X ,then the expected value of X will be shifted by that same constant amount.

ANSWER: D10. If 1 2 3, , X X X are three independent random variables with variances of 2,4, and 5,

respectively, then 1 2 3(2 3 4 )V X X X + + is

A. 32B. 12C. 108D. 140E. None of the above answers are correct

ANSWER: E

11. Which of the following statements is an example of a continuous random variable?

A. The depth of Lake Michigan at a randomly chosen point on the surface

B. The number of gas stations in DetroitC. The number of credit hours you have this semester D. All of the above

ANSWER: A

12. If the probability density function of a continuous random variable X is

( ) f x =.5 0 20 otherwise

x x

then, (1 1.5) P x is

A. .5625

B. .3125C. .1250D. .4375

ANSWER: B

13. A continuous random variable X is uniformly distributed on the interval [35, 45].The probability that X is between 40 and 50 is

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A. .1B. .7C. .6D. .5

ANSWER: D

14. Let X be a continuous random variable with probability density function f ( x) andcumulative distribution function F ( x). Then for any two numbersa and b witha = C. ( ) ( ) /( ) F x x a b a= D. ( ) ( ) 1 P X b F b> =

ANSWER: B

15. Which of the following is true about the median of a continuous distribution?

A. is the 50th percentileB. is the 75th percentileC. The area under the density curve to the right of is larger than the area to the

left of D. satisfies ( ) 1 F =%

ANSWER: A

16. A continuous distribution whose probability density function f ( x) is symmetric hasmedian equal to

A. 0B. 1C. f (.5)D. the value of x that is the point of symmetry

ANSWER: D

17. If X has a normal distribution with mean and standard deviation , and Z is thestandard normal random variable whose cumulative distribution function is

( ) ( ) P Z z z = , then which of the following statements are not correct?

A. ( ) / Z X =

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B. ( ) [( ) / ] [ ) / ] P a X b b a = C. ( ) 1 [( ) / ] P X a a = D. ( ) 1 [( ) / ] P X b b = E. All of the above statements are not correct

ANSWER: C18. If X is a normally distributed random variable with a mean of 80 and a standard

deviation of 12, then the probability that X = 68 is

A. .1587B. .0000C. .6587D. .8413E. None of the above answers are correct

ANSWER: B19. If X is a nonnegative random variable and the random variableY = ln( X ) is

normally distributed with parametersand , then which of the followingstatements are not true?

A. X is a lognormal random variable.B. The parameters and are the mean and standard deviation of X .C. The parameters and are the mean and standard deviation of Y.D. The cumulative distribution function (cdf) of X can be expressed in terms of

the cdf ( ) z of a standard normal random variable Z .

ANSWER: B

20. If X and Y are independent random variables with

(0) .5, (1) .3, (2) .2 and (0) .6, (1) .1, (2) .25, and (3) .05. Then ( 1 and 1) X X X Y Y Y Y p p p p p p p P X Y = = = = = = = A. .30B. .56C. .70D. .80

ANSWER: B

21. Which of the following statements are correct if a and c are either both positive

A. Corr( aX +b , cY +d ) = ab Corr( X ,Y )B. Corr( aX +b , cY +d ) = Corr( aX ,cY )C. Corr( aX +b , cY +d ) = Corr( X ,Y )

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D. Corr( aX +b , cY +d ) = ab Corr( X ,Y ) + bdE. None of the above statements is correct

ANSWER: C

22. Which of the following statements are correct for any two random variables X andY ?

A. 1 < Corr( X ,Y ) < 1B. 1 Corr( , ) 1 X Y C. Corr ( , ) 1 X Y D. Corr ( , ) 1 X Y E. 0


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ANSWER: A

Problem 3 (12%). A marine insurance company offers its policyholders a number of

different payment options. For a randomly selected policyholder, let X = thenumber of months between successive payments. The cdf of X is as follows:

0 1.30 1 3.40 3 4

( ).45 4 6.60 6 121 12

x

x

x F x

x

x

x

< < = = =

Problem 4 (12%). Suppose the number X of hurricanes observed in the Caribbeanduring a 1-year period has a Poisson distribution with9. = Using the table provided,

a. Compute ( 5). P X (2%) b. Compute (6 9). P X (3%)c. Compute (10 ). P X (3%)d. How many can be expected to be observed during the 1-year period? What is

the standard deviation of the number observed? (2%)e. Suppose we observe the region this year and no hurricanes occur. Given that

info, how many do you expect to occur in 2008? (1%)

x 1 3 4 6 12 P ( x) .30 .10 .05 .15 .40

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f. Suppose we observe the region this year and 23 hurricanes occur. Given thatinfo, how many do you expect to occur in 2008? (1%)

ANSWER:a. ( 5) (5;9) .116 P X F = =

b. (6 9) (9;9) (5;9) .587 .116 .471 P X F F = = =c. ( 10) 1 ( 9) 1 (9,9) 1 .587 .413 P X P X F = = = =d. ( ) 9, 3 x E X = = = =e. 9 hurricanes, because of memoryless property of the exponential distributionf. 9 hurricanes

Problem 5. (The Golden Oldie: a variation of last years problem 5!), 11%

NA387(3) has 31 students in 2007. The professor knows that the time needed to grade arandomly chosen second midterm exam is a uniform random variable with a mean of 15min and a rather small standard deviation of (square root of two) minutes.

a. If grading times are independent random variables, and the instructor beginsgrading at 3:45 PM, and grades continuously, what is the (approximate) probability that he is through grading before the 11:35 PM Late Show begins?(6%)

b. If the shows top ten list starts at 11:55 PM sharp, what is the (approx.) probability that the professor misses part of the list, assuming he waits untilgrading is finished before turning on the TV? (5%)

ANSWER:

E(T) = (31)(15) = 465 min

= n = 2 31 = 7.874 min

a. P(T 470) = P( Z (470-465)/7.784) = (0.64) = 0.7389

b. P(T > 490) = 1 P(T 490) = 1 - P( Z (490-465)/7.784) =1- (3.21) = 0.0007

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NA387exam2-07.SOLUTIONS Pmf& Cdf Question