STAT 311, Fall 2015 Name ________________________________________

Discussion Section: Please circle one!

LEC 001 TR 11:00AM-12:15PM FISCHER, ISMOR LEC 002 TR 9:30-10:45AM FISCHER, ISMOR

DIS 311 W 1:20-2:10PM Zhang, Yilin DIS 321 W 1:20-2:10PM Li, Xiaomao

DIS 312 W 12:05-12:55PM Zhou, Hao DIS 322 W 12:05-12:55PM Zhang, Yilin

FINAL EXAM PLEASE SHOW ALL WORK!

Problem Points Grade

1

20

2

60

3

35

4 10

5 25

Total

150

1. The arrival time of Joe’s usual morning bus (B) is normally distributed, with a mean ETA at 8:00 AM, and a standard deviation of 4 minutes. Joe’s arrival time (A) at the bus stop is also normally distributed, with a mean ETA at 7:50 AM, and a standard deviation of 3 minutes.

(a) Assuming independence between them, what is the distribution of the random variable X = A – B, the difference between the two arrival times? Show all work. (5 pts)

(b) In order to be able to catch the bus, Joe must arrive at the bus stop before the bus does. With what probability will this occur? (Hint: What must be true about X?) Show all work.

(7 pts)

(c) On average, how much earlier should Joe arrive, if he expects to catch the bus with 99% probability? (That is, what should his new “mean ETA” be?) Show all work. (8 pts)

2. Part of a productivity study conducted by a certain agency involves determining the coffee-drinking habits of its employee population. The following joint probability mass function (pmf) ( , )f x y for the number of cups of coffee consumed in the morning (X) and afternoon (Y) is eventually obtained, and expressed in tabular form:

X = AM # cups 0 1 2

Y =

PM

#

cups

0 0 .10 .15

1 .20 .25 .30

(a) Calculate and label the respective marginal pmfs ( )Xf x and ( )Yf y on the table above. (7 pts) (b) Are X and Y statistically independent? Provide a formal reason below. (4 pts)

(c) Calculate the mean [ ]X E Xµ = and variance 2 ( )X Var Xσ = . Show all work. (7 pts)

(d) Calculate the mean [ ]Y E Yµ = and variance 2 ( )Y Var Yσ = . Show all work. (7 pts)

(e) Calculate the covariance ( , )XY Cov X Yσ = between X and Y. Show all work. (7 pts)

(f) Calculate the correlation coefficient ( , )XY Corr X Yρ = between X and Y, and classify their association as either positive or negative, and either strong, moderate, or weak. Show all work.

(7 pts)

(g) Construct a probability table for X + Y = the total daily number of cups consumed. Show all work. (7 pts)

(h) Calculate its mean and variance directly from this table. Show all work. (7 pts)

(i) Use the previous results to confirm that the formulas below hold in this example. Show all work.

• [ ] [ ] [ ]E X Y E X E Y+ = + (2 pts)

• ( ) ( ) ( ) 2 ( , )Var X Y Var X Var Y Cov X Y+ = + + (5 pts)

3. An important special case of the standard Beta distribution is the power distribution on [0, 1]. For any real number 1r > , define the probability density function (pdf) for a continuous random variable T = “Time to Failure” of a system component via the following.

1, 0 1( )0, otherwise

r

Tr t tf t

− ≤ ≤=

(a) Explicitly show that this is a legitimate pdf. Show all work. (4 pts)

(b) Determine the cumulative distribution function (cdf) ( ) ( )F t P T t= ≤ . Show all work. (4 pts)

(c) Determine the expected lifetime of a system component having this power density. Show all work. (4 pts)

Suppose a certain system has two independent components, whose lifetimes X and Y are modeled by the pdfs 1( ) p

Xf x p x −= if 0 1x≤ ≤ (and 0 elsewhere) and 1 ( ) qYf y q y −= if

0 1y≤ ≤ (and 0 elsewhere), for some parameters 1p > and 1q > , respectively.

(d) Determine the joint probability density function ( , )f x y over the unit square in the XY-plane. (3 pts)

(e) Determine the expected mean failure time, i.e., the expected value of .2

X Y+ [Hint: Use (c).]

Show all work; formally justify your steps. (4 pts)

(f) Determine the probability that both components fail before halfway through [0, 1], i.e., [ ]1 1

2 2 .P X Y< < Show all work. (4 pts)

(g) Determine the probability that at least one component fails before halfway through [0, 1], i.e., [ ]1 1

2 2 .P X Y< < Show all work. (4 pts)

(h) Determine the probability that component Y fails before component X fails, i.e., ( ).P Y X< [Hint: Sketch the area in the XY-plane.] Show all work. (8 pts)

4. (a) Recall that the covariance between any two random variables X and Y is defined by

[ ]( , ) ( )( ) .X YCov X Y E X Yµ µ= − −

Prove that an equivalent, alternate form is

( , ) [ ] .X YCov X Y E XY µ µ= − Show all work; justify all steps. (8 pts)

(b) Use the result of part (a) to derive the equivalent, alternate form for the variance of a random variable X, defined by

2( ) ( ) .XVar X E X µ = −

(2 pts)

5. Bob suddenly remembers that today is his girlfriend’s birthday, and rushes into a nearby florist to buy her some flowers. There he finds a large population of different kinds of roses, whose individual costs (X) are uniformly distributed between $1 and $5 apiece, as shown.

As he is in a hurry, he simply intends to independently select a dozen roses at random, and bring them up to the counter to pay the total cost 1 2 12Y X X X= + + + .

(a) Calculate the theoretically lowest and highest possible total costs that Bob could pay for a dozen roses selected at random. Explain. (5 pts)

(b) How much money can Bob expect to pay for a dozen roses (i.e., on average)? Explain. (5 pts)

(c) Calculate the standard deviation for the cost of a single rose. Show all work. (5 pts)

(d) What is the approximate probability that he will have to pay between $30 and $45? (Hint: First apply the Central Limit Theorem to find the sampling distribution of .X Then determine how X and Y are related.) Show all work. (10 pts)

( ) 1 4f x =