112 CHAPTER 2 Probability

Exercises for Section 2.4

1. Detennine whether each of the following random vari­ables is discrete or continuous.

a. The number of heads in 100 tosses of a coin.

b. The length of a rod randomly chosen from a day's production.

c. The final exam score of a randomly chosen stu­dent from last semester's engineering statistics class.

d. The age of a randomly chosen Colorado School of Mines student.

e. The age that a randomly chosen Colorado School of Mines student will be on his or her next birthday.

2. Computer chips often contain surface imperfections. For a certain type of computer chip, the probability mass function of the number ofdefects X is presented in the following table.

a. Find P(X 2).

b. Find P(X > 1).

c. Find f.Ox.

d. Find

3. A chemical supply company ships a certain sol­vent in 10-gallon drums. Let X represent the num­ber of drums ordered by a randomly chosen cus­tomer. Assume X has the following probability mass function:

a. Find the mean number of drums ordered.

b. Find the variance of the number of drums ordered.

c. Find the standard deviation ofthe number ofdrums ordered.

d. Let Y be the number of gallons ordered. Find the probability mass function of Y.

e. Find the mean number of gallons ordered.

f. Find the variance of the number of gallons ordered.

g. Find the standard deviation of the number of gal­lons ordered.

4. Let X represent the number of tires with low air pres­sure on a randomly chosen car.

a. Which of the three functions below is a possible probability mass function of X? Explain.

x 0 1

0.2 0.2 0.3 0.1 0.1PI(X) 0.1 0.3 0.3 0.2 0.2p:c(x) 0.1 0.2 0.4 0.2 0.1P3(X)

b. For the possible probability mass function, com­pute f.Ox and a~.

5. A survey ofcars on a certain stretch ofhighway morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represent the number of occupants in a randomly chosen car.

a. Find the probability mass function of X.

b. Find P(X ::S 2).

c. Find P(X > 3).

d. Find f.Ox.

e. Find ax.

6. The element titanium has five stable occurring iso­topes, differing from each other in the number of neu­trons an atom contains. If X is the number of neutrons in a randomly chosen titanium atom, the probability mass function of X is given as follows:

x 24 25 26 27 28 p(X) 0.0825 0.0744 0.7372 0.0541 0.0518

a. Find f.Ox.

b. Find ax.

7. A certain type of component is packaged in lots of four. Let X represent the number of properly turlctlommg components in a randomly chosen lot.

.Assume that the probabi Iity that exac tly x components functi on is proportional to x; in o ther words, assume

mat the probability mass func tion of X is given by

ex x = 1, 2 , 3. or 4 p (x) = 0{ otherwise

v. here e is a constant.

Find the value of the consta nt c so that p(x) is a

probability mass functio n.

b. Find P(X = 2).

Find the mean number of properly func tioning

components .

d. Find the variance of the number of properly func ­

tioning components.

e. Find the standard deviation of the number of prop­erly func tioning components.

' . After manu fac ture, computer di sks are tes ted for er­rors. Let X be the number of errors detec ted on a

randoml y chosen di sk. The fo llowing table presents values of the cumul ative dis tribution function F (x)

of X.

x F (x)

0 OA I 0. 72

2 0.83 3 0.95 4 LOO

a. What is the probability tha t two or fewer errors are de tected?

b. What is the probability that more thall three en-ors are detec ted ?

c. What is the probability that exac tly o ne error is detected ?

d. What is the probability th at no errors are detec ted ?

e. What is the most probable number of elTors to be detec ted ?

On 100 different days, a traffic eng ineer counts the number of cars that pass tlu'ough a ce11ain in tersection between 5 P.M. and 5:05 P.M . The results are presented

in the following table.

2.4 Random Variables 113

Number Number Proportion of Cars of Days of Days

0 36 0.36 1 28 0.28 2 15 0.15 3 10 0. 10 4 7 0.D7 5 4 0.04

a. Let X be the number of cars pass ing tlu'ough the inte rsec tion between 5 P.M. and 5:05 P.M. on a ran­

domly chosen day. Someone sugges ts that for any positive integer x, the probability mass func tion

of X is P, (x) = (0.2)(0.8)' . Using th is fun c tion,

compute P(X = x) for values of x from 0 throu gh 5 InclUSive.

b. Someone e lse sugges ts that fo r any positive inte ­

ger x, the probability mass function is /h (x ) = (0 .4)(0.6)'. Using thi s func tion, compute P (X = x) fo r va lues of x from 0 through 5 inc lu sive.

c. Compare the resu lts o f parts (a) and (b) to the da ta in the table. Which probability mass func ti on ap­

pears to be the be tte r model? Explain .

d. Someone says that neither of the fun ctions is a

good model si nce neither one agrees with the data exactl y. Is th is right" Expla in .

lO. Microprocessing chips are randomly sampled one

by one from a large po pUlation, and tes ted to de ­te rmi ne if they are acceptable for a certain applica­

tion. Ninety percent of the ch ips in the popul ation are

acceptable.

a. What is the probability that the fi rst chip chosen is

acceptable?

b. What is the probability that the first chip is unac­ceptable, and the second is acceptable?

c. Le t X represent the number of chips that a re tes ted

up to and including the firs t acceptable chip . Find

P(X = 3) .

d. Find the probability mass func tion of X.

11. Refer to Exercise 10. Let Y be the number of chips

tes ted up to and including the second acceptable chip.

a. What is the smallest possible va lue for y.)

b. What is the probability that Y takes 0 11 that value ?

114 CHAPTER 2 Probability

c. Let X represent the number of chips that are tested 3. What proportion of steel plates have elongations up to and including the first acceptable ch ip. Find greater than 25 %? PlY = 31X = I). b. Find the mean elongation.

d. Find PlY = 31X = 2). c. Find the variance of the elongations. e. Find PlY = 3). d. Find the standard deviation of the elongations.

e. Find the cumu lative distribution function of the12. Three components are randomly sampled, one at a elongations. time, from a large lot. As each component is selected ,

it is tested . If it passes the test, a success (S) occurs: if f. A particular plate elongates 28% . What proportion

it fails the test, a failure (F) occurs. Assume that 80% of plates elongate more than this?

of the components in the lot will succeed in passing the 15. The lifetime of a transistor in a certain appl ication

test. Let X represent the number of successes among has a lifetime that is random wi th probability density

the three sampled components. fun ction

a. What are the possible values for X'J - 0 I I 0 I 1>0 b Find P(X = 3). f(t) = 0 e .

{ 1:'00 c. The event that the first (;omponent fails and the next

two succeed is denoted by FSS. Find P (FSS). a. Find the mean lifetime.

d. Find P(SFS) and P (SSF). b. Find the standard deviation of the lifetimes.

e. Use the results of parts (c) and (d) to find c. Find the cumulative distribution function of the P(X = 2). lifetime.

f. Find P(X = I) . d. Find the probability that the lifetime will be less than 12 months.g. Find P ( X = 0) .

h. Find /J.. x . 16. A process that manufactures piston rings produces I. Find a.~. rings whose diameters (in centimeters) vary according

J. Let Y represent the number of successes if four to the probability density function

components are sa mpled. Find PlY = 3). 9.75 <x < 10.25 f(x) = {~[l - 16(x - lW] otherwise

between 80 Q and 120 Q. Let X be the mass of a ran­13. Resistors labeled 100 Q have true resistances that are

a. Find the mean diameter of rings manufactured bydoml y chosen resistor. The probability density func­this process. tion of X is given by

b. Find the standard deviation of the diameters of X - 80

80 < x < 120 rings manufactu red by this process. (Hill l: Equa­f(x) = 8~0 tion 2.36 may be easier to use than Equation 2.37 .)

{ otherwise c. Find the cu mulative distribut ion function of piston

a. What propOition of resistors have resistances less ring diameters. than 90 Q? d. What propOition of pis ton rings have diameters less

b. Find the mean resistance. than 9.75 cm?

c. Find the standard deviation of the resistances. e. What proportion of piston rings have diameters between 9.75 and 10.25 cm?d. Find the cumulative distribution function of the

resistances. 17. Refer to Exercise 16. A competing process produces rings whose diameters (in centimeters) vary according 14. Elongation (in percent) of steel plates treated wi th alu­to the probability density function minum are random with probability densi ty function

15[1 - 25(x - 10.05)']/420 < x < 30

f(x) = 985 < x < 10.25{otherwise o otherwise

Spec ifications call for the diameter to be 10.0±0.1 cm. Which process is better, this one or the one in Exer­

cise 16? Explain.

The lifetime, in years, of a certain type of fuel cell is

a random variable with probability density function

81 x > 0

f(x) = (x ~ 3)"{ ,x ::: 0

a. What IS the probability that a fuel cell lasts more than 3 years?

b. What is the probabi Iity that a fuel cell lasts between I and 3 years?

c. Find tbe mean lifetime.

d. Find the vari ance of the lifetimes.

e . Find the cumulative distribution function of the

lifetilne.

f. Find the median lifetime.

g. Find the 30th percentile of the lifetimes.

The level of impurity (in percent) in the product of a certain chemical process is a random variable with

probability density function

3 0

0 < x < 464 X-~4 - x) { otherwise

a. What is the probability that the impurity level is

greater than 3%?

b. What is the probability that the impurity level is

between 2% and 3%°

c. Find the mean impurity level.

d. Find the variance of the impurity level s.

e. Find the cumulative distribution function of the

impurity leve l.

The main bearing clearance (in mm) in a certain type

of engine is a random variable with probability density

runction

625.1 o < x ::: 0 .04

f(x) = 50 - 0625X 0.04 < x ::: 0. 08 {

otherwi se

a. What is the probability that the clearance is less

than 0.02 mm?

b. Find the mean clearance.

c . Find the standard deviation of the clearances.

2,4 Random Variables 115

d. Find the cumulative distribution function of the clearance.

e. Find the median clearance.

f. The specification for the clearance is 0.015 to

0.063 mm. What is the probability that the speci­fication is met?

21. The concentration of a reactant is a random va riable

with probability density function

O< x < 1f(x) = {1.2(X + x") o othelwise

a. What is the probability that the concentration is greater than 0.5?

b. Find the mean concentration .

c, Find the probability that the concentration is within

±O. I of the mean.

d. Find the standard deviation a of the concentrations.

e. Find the probability that the concentration is within

±2a of the mean.

f. Find the cumu lative di stribution function of the concentration,

22. The error in the length of a part (absolute value of

the di fference between the actual length and the target

length). in mm, is a random variable with probabil ity density functi on

O<x<1 f (x) = { I ~::- I

otherwise

a. What is the probability that the error is less than 0.2 mm ?

b. Find the mean error.

c. Find the variance of the error.

d. Find the cumulative distribution functi on of the

error.

e . Find the median error.

f. The specification for the elTor is 0 to 0 .3 mm. What is the probability that the specification is met?

23. The thickness of a washer (in rrun) is a random vari­able with probability density functi on'

2 <..1'<4f( x) = x){ :2X(~ ­othelwise

116 CHAPTER 2 Probability

a. What is the probability that the thickness is less 25. The repair time (in hours) for a certain machine is a than 2.5 m? random variable wi th probability density function

b. What is the probability that the thickness is between 2.5 and 3.5 m? x x>Of( x) _ { xe ­- 0c. Find the mean thickness. x.:sO

d. Find the standard deviation a of the thicknesses.

e. Find the probability that the thickness is wi thin ±a a. What is the probability that the repair time is less of the mean. than 2 hours?

f. Find the cumulative distribution function of the b. What is the probability that the repair time is be­thickness. tween 1.5 and 3 hours?

c. Find the mean repair time. 24. Particles are a major component of air pollution in many areas. It is of interest to s tudy the sizes of con­ d . Find the cumulative distribution function of the

taminating particles. Let X represent the diameter, in repair times .

micrometers, of a randomly chosen particle. Assume 26. The diameter of a rivet (in mm) is a random variable

that in a certain area, the probability density function with probability density function

of X is inversely proportional to the volume of the particle; that is, assume that

6(X - 12)(13 - x) 12 < x .:s 13 f(x) = 0

x~l { otherwise

f(x) = { ~1 x < 1

a. What is the probability that the diameter is less where c is a constant. than 12.5 mm?

a. Find the value of c so that f(x) is a probability b Find the mean diameter.

density function. c. Find the standard deviation of the diameters.

b. Find the mean particle diameter. d. Find the cumulative dist.ribution function of the

c. Find the cumulative distribution function of the diameter.

particle diameter. e. The speci fi cation for the diameter is 12.3 to

d. Find the median particle diameter. 12.7 mm. What is the probability that the speci­fication is met?e. The term PM IO refers to particles 10 l.1.m or less

in diameter. What proportion of the contaminating particles are PM 10 ?

f. The term PM2.5 refers to particles 2.5 I1-m or less in diameter. What proportion of the contaminating particles are PMz5 ?

g. What proportion of the PM 10 particles are PMz)?

2.5 Linear Functions of Random Variables

In practice we often construct new random variables by performing arithmetic operations on other random variables. For example, we might add a constant to a random variable, multiply a random variable by a constant, or add two or more random variables together. In this section, we describe how to compute means and variances of random variables