A Very Nice Practice Midterm Exam

A representative sampling of the first half of prob and stat- Complete with follow-on questions to enhance the learning process

Need to know in general sample spaces and random events

Unions, intersections, and complements counting methods

Combinations, permutations, fundamental counting rule condition probabilities, total prob rule, & Bayes

theorem discrete random variables

Probabilities, means and variances continuous random variables

Probabilities, means and variances, modes and medians joint distributions

covariance and correlation marginal and conditional distributions

Descriptive statistics

Question 1The chief executive officer (CEO) of the Combinatorial Company, Mr. Hi N. Mitey, will be visiting 3 of his 5 plants this week. The plant manager of the Homestead Plant is betting that the CEO will not visit him this week and is therefore delaying preparations for his visit. If Hi N. Mitely selects his itinerary randomly, what is the probability that the Homestead Plant will not be visited this week?

4 1

3 0favorable 4Prob

5total 10

3

order of visits is not important

Alternate approach

4 3 2favorable 2

Probtotal 5 4 3 5

Follow-up to Question 1 The chief executive officer (CEO) of the

Combinatorial Company decides not to visit the plants. Instead he will send his 3 vice-presidents. Each VP selects a plant at random to visit. What is the probability that the Homestead Plant is not selected?

3

3

favorable 4 64Prob .512

total 5 125

Bonus Question #1

Faculty advisors are assigned randomly to new students entering the Engineering Management program from among five full-time faculty members. Three new students are admitted on a particular day. The office staff member, Dizzy Dunce, assigns (randomly) each student a faculty advisor. (a) What is the probability that each student is assigned to a different advisor? (b) What is the probability that all three students are assigned to one faculty member?

(a) (5)(4)(3) / 53 = .48

(b) 5 / 53 = 1/25 = .04 (5 ways to select the same faculty member)

Question 2 Eighty percent of all students taking probability and

statistics pass the course while seventy-five percent of those taking operations research pass the course. However, of those that have passed probability and statistics, 90 percent will pass operations research. What fraction of students passes both courses? What fraction will pass at least one course?

Let A = the event, student passes prob/stat B = the event, student passes operations researchGiven: P(A) = .80; P(B) = .75; P(B|A) = .90Required: P(A B) = ? and P(A B) = ?

P(A B) = P(B|A) P(A) = (.90) (.80) = .72

P(A B) = P(A) + P(B) - P(A B) = .80 + .75 - .72 = .83

Question 3 The Rockweed Aircraft Corporation is redesigning the

instrument panel for its new F-222 steam propelled fighter. After interviewing a large number of pilots, they decided to

replace certain gauges with idiot (warning) lights. One such light warns of engine trouble. There is a probability

of .01 that there will be engine trouble for a single mission of this aircraft. Given there is engine (boiler) trouble, there is a .99 probability

that the idiot light turns on. If there is no engine trouble during the mission, there is a .98

probability that the idiot light will not turn on. What is the probability that the engine light turns on during a

mission. If the engine light turns on during a mission, what is the

probability of engine trouble?

Question 3 (continued)Let T = the event, engine trouble O = the event, engine light turns onGiven: P(T) = .01; P(O|T) = .99; P(Oc|Tc) = .98

The Total Probability Rule (TPR):

P(O) = P(O|T) P(T) + P(O|Tc) P(Tc)P(O) = .99 (.01) + .02 (.99) = .0297

Then Bayes:

P(T|O) = P(O|T) P(T) / P(O) = .99 (.01) / .0297 = .333

Required: P(O) and P(T|O)

Question 4The table summarizes the GPA and statistics aptitude test results

of students admitted to the grad engineering programs. Based upon these historical numbers, determine the relative

frequency probability that an individual applying for a grad engineering program:

(a) will score low on the aptitude test and have a GPA above 3.5(b) will score high on the aptitude test if individual has a GPA

between 3.0 and 3.5(c) Show that test scores and GPA are not independent events

Entering Grade Point Average (GPA)

Below 3.0 3.0 – 3.5 Above 3.5

Aptitude test results

High 0 17 13

Medium 5 40 5

Low 15 3 2

Question 4(a) will score low on the aptitude test and have a GPA above 3.5(b) will score high on the aptitude test if individual has a GPA

between 3.0 and 3.5(c) Show that test scores and GPA are not independent events

Entering Grade Point Average (GPA)

Below 3.0 3.0 – 3.5 Above 3.5

Aptitude test results

High 0 17 13

Medium 5 40 5

Low 15 3 2

total 20 60 20

(a) P(score low and GPA > 3.5) = 2/100(b) P(score high | 3<GPA<3.5) = 17/60(c) P(High|GPA<3) = 0 P(High) = 30/100

30

50

20

Question 5 The probability that a missile fired at a terrorist target will

destroy the target is .6. Assuming independence, what is the minimum number of missiles to be launched against the target in order to have at least a 90 percent chance of destroying the target?

Let X = the discrete random variable, the number of missiles fired untilthe target is destroyed.

X ~ Geo(.6)

1

4

5

3

1

2

Pr{

( ) .6 .4 , ( ) 1 .4

Pr{ 1} (1) 1 .4 .60

Pr{ 2} (2) 1 .

3} (3)

4 .84

Pr{ 4} (4) 1 .4 .9744

Pr{ 5}

1 .4 .936

(5) 1 .4 .98976

x xf x F x

X F

X F

X F

X

X

F

F

Follow-on Question 5 The probability that a missile fired at a terrorist target will

destroy the target is .6. Assuming independence, what is the probability of destroying the target with the 3rd missile? What is the expected number of missiles needed to destroy the target?

Let X = the discrete random variable, the number of missiles fired untilthe target is destroyed.

X ~ Geo(.6)

1

3 1

( ) .6 .4 , ( ) 1 .4

Pr{ 3} (3) .6 .4 .096

1[ ] 1.667

.6

x xf x F x

X f

E x

Bonus Question #2 The Catastrophic Construction Company provides emergency

repair of facilities damaged during natural disasters such as hurricanes and tornados. They respond primary to requests from FEMA (Federal Emergency Management Agency). The number of such requests per year is a random variable best described by the following probability mass function (PMF):

24

( ) , 0,1,2,3,4,531

xf x x

E[X] = 0 (16/31) + 1(9/31) + 2(4/31) + 3(1/31) + 4(0/31) + 5(1/31) = 25/31 = .806 E[X2] = 0 (16/31) + 1(9/31) + 4(4/31) + 9(1/31) + 16(0/31) + 25(1/31) = 1.903V[X] = 1.903 - .8062 = 1.253, = 1.12

Question 6The Loose Screw Company manufactures nuts and bolts

for sale to wholesale distributors. The delivery time (lead-time) for orders in days is given by the following Probability Density Function (PDF):

f(x) = x/300 , 5 x 25

Find the probability that delivery will be within one standard deviation of the mean.

note correction

Question 62 2

5 5

2525 2 3

5 5

2525 3 42

5 5

2 2

25( ) , 5 25; ( )

300 300 600 600

15625 125[ ] 17.222

300 900 900

390625 625[ ] 325

300 1200 1200

325 17.222 28.402, 5.3294

Pr{17.222 5.3294 17.222 5.329

xxx z z xf x x F x dz

x xE X dx

x xE X dx

x

2 2

4} Pr{11.89 22.55}

22.55 25 11.89 25(22.55) (11.89) .80584 .19395 .612

600 600

x

F F

The Geometry of Question 625/300

5/300

5 25x

f(x)

f(x) = x/300 , 5 x 25

Area = (20)(5/300) + (1/2)(20)(20/300) =1

20/300

x

2

5 1 5( ) 5 5

300 2 300

(5)(2) 5 5 5 25

600 600

xF x x x

x x x x

Question 7 Given the following joint probability mass

function, find the correlation between X and Y.

X/Y 0 1 2

0 .1 .3 .2

1 .2 .1 .1

y 0 1 2f(y) .3 .4 .3E[Y] = 0 (.3) + 1(.4) + 2(.3) = 1.0V[Y] = 0 (.3) + 1(.4) + 4(.3) - 12 = .6

X/Y 0 1 2 totals E[X] E[X^2}

0 0.1 0.3 0.2 0.6 0.4 0.4

1 0.2 0.1 0.1 0.4 V[X] = 0.24totals 0.3 0.4 0.3

E[Y] = 1 E[Y^2] = 1.6 V[Y] = 0.6E[XY] = 0.3

covar = -0.1 Cor[XY] = -0.263523

Question 8 Ms. Ima Borne Loser makes correct decisions 10 percent of

the time. How many (independent) decisions must she make to have at least a 90 percent chance of making at least one correct decision? Hint: let Ei = the event, the ith decision is correct.

1 2

1 2 1 2

1

Given: ( ) .10, therefore ( ) .9

Required : ( ... ) .9, find

( ... ) 1 ( ... )

1 ( ) ( ) 1 .9 .9

.9 .1

ln .1ln .9 ln .1 21.85; 22

ln .9

ci i

n

c c cn n

c c nn

n

P E P E

P E E E n

P E E E P E E E

P E P E

n or n n

Question 8 – alternate approach

Ms. Ima Borne Loser makes correct decisions 10 percent of the time. How many (independent) decisions must she make to have at least a 90 percent chance of making at least one correct decision?

Let X = a discrete RV, the number of correct decisions in n trials

( ,.1)

Pr 1 1 Pr 0 1 (1 .1) 1 .9 .9n n

X B n

X X

n Pr{X 1}10 .651315 .794120 .878425 .9282

n Pr{X 1}21 .890622 .9015

Question 9

Thirty percent of the Fly-By-Nite Airline Company’s flights are delayed. If Mr. I. N. Hurrie is scheduled on 5 flights as he travels to each of the company’s 4 technical centers, what is the probability of no more than one flight being delayed? Assume Independence.

Let X = a discrete RV, the number of delayed flights among the 5

X ~ B(5, .3)

Pr{X 1} = f(0) + f(1) = F(1) = .528 (from the Prob Calculator)

What is the expected number of delayed flights? E[X] = (5)(.3) = 1.5

Question 10 Given the following CDF, find the mean of the

probability distribution.

2

( ) , 0 10100

xF x x

1010 3

0 0

( )( )

50

1000[ ] 6.67

50 150 150

dF x xf x

dx

x xE X x dx

Follow-on Question 10 Given the following CDF, find the variance of

the probability distribution.

2

( ) , 0 10100

xF x x

1010 42 2

0 0

2

10000[ ] 50

50 200 200

[ ] 50 6.6667 5.55511; 2.3569

x xE X x dx

V X

Follow-on to the Follow-on of Question 10

Given the following CDF, find the mean and variance of the probability distribution.

2

( ) , 0 10; ( )100 50

x xF x x f x Isn’t this just a right

triangular distribution with b = 10?

2

2

2

2( ) 0

( )

2[ ]

3

xf x x b

b

xF x

b

E X b

Var[X] = 2.3572

from Prob CalcBeta ( =2 =1)

Question 11

Professor I. Do Little’s research (NSF Grant) on the migration pattern of the coastal plain swamp sparrow has determined that their migration (flying) time is normally distributed with a mean of 23 days and a variance of 36 days. Compute the probability that a particular swamp sparrow will migrate between 30 to 36 days.

(23,36)

Pr{30 35} (35) (30)

35 23 30 23Pr Pr Pr 2 Pr 1.17

6 6

.9772 .8783 .0989

X n

X F F

z z z z

Follow-on Question 11

Professor I. Do Little’s research (NSF Grant) on the migration pattern of the coastal plain swamp sparrow has determined that their migration (flying) time is normally distributed with a mean of 23 days and a variance of 36 days.

90 percent of the sparrow population will complete their migration in how many days?

(23,36)

Pr{ } .90

23Pr .90; 1.2816

6

23 1.2816 6 30.6896 days

X n

X x

xz z

x z

Question 12

Ted E. Bare, an engineering student, has observed that the number of times the campus police check the parking lot has a Poisson distribution with a mean of once every 4 hours. If Ted E. is parked illegally during a 3-hour evening class, what is the probability he will get ticketed?

0.75.75

( 3 .25)

.75Pr 1 1 (0) 1 1 .5276

0!

X Pois x

eX f e

Let X = a discrete RV, the number of times in a 3-hour period, theCampus police will check the parking lot.

Follow-on to Question 12 At break time, one and half hours into the class, Ted checks

and sees that he did not receive a ticket as yet. What is the probability that he will get ticketed by the time class is out?

0.375.375

( 1.5 .25)

.375Pr 0 .6873

0!

X Pois x

eX e

Also, let Y = a continuous RV, the time to the next arrival of the campus policeY ~ Exp( = .25) and F(y) = 1 – e-.25y

Therefore Pr{Y<1.5} = 1 – F(1.5) = 1 - e-.25(1.5) = .3127

Let X = a discrete RV, the number of times in a 1.5-hour period, theCampus police will check the parking lot.

Prob get ticket= 1 - .6873 = .3127

Question 13 Dawn E. Brook is an undergraduate student who works part-

time in the Engineering Management office. Among her duties is to make coffee for the faculty and staff. The time it takes the old coffee pot to make coffee is a random variable having a Weibull distribution with = 2 and = 20 minutes. If the coffee pot perks for less than 16 minutes, the coffee will be too weak and if the coffee pot perks for more than 23 minutes, the coffee will be too strong. What is the probability that Dawn E. Brook will brew a pot of coffee to the faculty’s liking?

(2, 20)

Pr 16 23 (23) (16) .73353 .47271 .261

X Weib

X F F

Alternate Question 13 Dawn E. Brook is an undergraduate student who works part-

time in to the Engineering Management office. Among her duties is to make coffee for the faculty and staff. The time it takes the old coffee pot to make coffee is a random variable having the following cumulative distribution (CDF):

If the coffee pot perks for less than 16 minutes, the coffee will be too weak and if the coffee pot perks for more than 23 minutes, the coffee will be too strong. What is the probability that Dawn E. Brook will brew a pot of coffee to the faculty’s liking?

What is the mean brewing time?

(23) (16) .73353 .47271 .261F F

2

20( ) 1 , 0x

F x e x

Weib(2,20)

[ ] 17.72min. from ProbCalc

X

E X

Question 14 Laye Z. Jones has been given three tasks to complete within

the next 8 hours. The time for Laye Z. to complete each task based upon past performance is normally distributed with the following parameters:

Task Mean Variance

1 3.4 hr. 2.4

2 4.8 hr. 3.8

3 2.5 hr. 1.8

What is the probability that Laye Z. will complete all 3 task on time.

1 2 3

(10.7,8)

8 10.7Pr 8 Pr Pr .9546 .1699

8

Y X X X

Y n

Y z z

Totals 10.7 8.0

Question 15 The time it takes to complete an engineering design project in

days is a random variable having the following probability density function (PDF): 2

6

3for 0 100 days

( ) 100 otherwise

xx

f x

The profit resulting from completion of the project is given by $5,000 / X. Find the expected profit.

100 1002

6 60 0

1002

6

0

5000 3 15000[Pr ]

10 10

15,000 15,000$75

2 10 2 100

xE ofit dx x dx

x

x

x x

100100 1002 43

6 6 60 0 0

3 3 3 300[ ] 75 days

10 10 4 10 4

x xE X x dx x dx

x

500066.67 75

75

Bonus Question #3 Given the following very fine probability density function (PDF)

where the random variable X is Professor Domkoff ’s driving time in hours to school, find the (a) mean and the (b) probability that the driving time is no more than 90 minutes.

2

2( ) , 1 2f x x

x

22

2 11

21 1

22ln 2ln 2 2ln1 1.3863 hr

2 2 2 2 2( ) 2

1

2(1.5) 2 .667

1.5

xx

xdx x

x

F x dzz z x x

F

Problem 16 – Descriptive Stats & point estimation

The following random sample was obtained by measuring the time in (working) hours to complete a particular construction job. Treating the data as continuous, answer the following questions:

(a) Find a sample estimate for the population mean (b) Find a sample estimate for the population variance(c) Is the data skewed left, right, or almost symmetrical?(d) Find the sample interquartile range.(e) Find the sample median.(f) Based only on a histogram, which of the following

distributions are not likely models for the population distribution?NormalWeibullRectangularExponentialTriangular

A Truly Great Set of Data

83.8 65.4 135 99.7 66.5 30.964.2 98.1 56.4 49.1 44.5 76.789.8 46.8 99.8 98.9 55.6 36.882.3 86.9 64.6 85.2 136 4890.4 71.8 95.7 110 91.7 71.563.3 56.2 85.3 68.440.4 72.1 75.8 123104 73.7 88.5 58.6108 77.2 71.7 74.1

96.6 113 72 67.9

Task Time in Hours

sample size 50mean 78.420variance 569.696std dev 23.868median 74.951st quartile 64.33rd quartile 95.7interquatile range 31.4Mimimum 30.9Maximum 135.7Range 104.8Skewness 0.3234Kurtosis -0.0294

Problem 16

From the histogram rule out the rectangular and exponential- also compare mean and std. dev.

some skewness to the right

0

2

4

6

8

10

12

14

16

18

47 64 81 98 115 132 149

sturges rule = 6.606601 6interval width = 17.46667 17

Keys to successful completion of the midterm

Work every problem if unable to complete – submit a partial or intermediate

answer state any assumptions, identify type of problem anything submitted correctly about problem will earn

some credit no credit for simply restating the problem

Apply common sense to your answers probabilities cannot exceed one, variances are positive does the magnitude of the number make sense

Use the computer and Prob Calculator saves time, less chance of an error

Start by defining events or RV’s write down what’s given follow up with what is required

Key Things to know

Recognize problem type random events versus random variables

Recognize types of random events dependent versus independent events conditional versus unconditional probability total probability problem (weighted average) Bayes problem

Recognize probability distributions theoretical density and distribution functions

i.e. Poisson or uniform PMF; exponential or Weibull PDF distinguish between binomial and geometric joint versus marginal versus condition distributions

The Important Distributions

The discrete ones Uniform Binomial Geometric Poisson

The continuous ones Rectangular Normal Exponential Weibull Triangular

Left triangular Right triangular

Key things to expect A combinatorial problem or an a priori probability

# favorable /total use of counting methods

A random event problem conditional or Bayes independent

A binomial, geometric or both A normal distribution problem A general distribution

find mean, median, variance, probabilities, median A joint distribution – most likely discrete

find marginal, conditional – probability, mean, variance, correlation

A problem requiring integration find a mean or probability