Homework Chapter 7 Random Variables

Random Variable ExamplesAP Statistics

. State whether each of the following random variables is discrete or continuous:

• The number of defective tires on a car. ÿ..>ÿ,,fÿ

• The body temperature of a hospita patient. L U]-ÿ,ll'ÿ- >

• The number of pages in a book. ÿ (/ÿ c., <ÿ-• The lifetime of a light bulb. ÿ ÿ ÿ ÿ ÿ lÿ ÿÿ

2. Let X be a number between 0 and 1 produced by a random number generator. Assumingthat the random variable X has a uniform distribution, find the following probabilities:

• P(X>.49) . ÿ--ÿ ÿ.- ÿ'1"1ÿ) I L_S• P(X > .49) ÿ c,-ÿ L.,<;{ ÿ" l)r,-. I

• P(0.19 < X < 0.37 or 0.84 < X <.ÿ7ÿ.

3. The Normal distribution with mean/.I = 6.8 and standard deviation a = 1.6 is a good

description of the Iowa Test of Basic Skills (ITBS) vocabulary scores of seventh-gradestudents in Gary, Indiana. Call the score of a randomly chosen student X for short. Find

P(X > 9) and interpret the result. Draw a graph and shade.

-2 /-7;T, . s

.,-, .ÿ' ,,( ,ÿ, #4 ,€,Vk{ÿ /"

4. A s of 12,000 able-bodied male students at the University of Illinois found tlÿat " ÿ.;their times for the mile run were approximately Normal with mean 7.11 minutes andstandard deviation 0.74 minute. Choose a student at random from this group and call his

time for the mile Y. Find P(Y < 6) and interpret the result. Draw a graph and shade.

, i,

P(x)

5. Let X be the number of courses for which a randomly selected student at a certain

university is registered. The probability distribution of random variable X appears inthe accompanying table.

X 1 2 3 4 5 6 7

.02 .05103 .09 .25 .gO .16

What is P(X = 4)? • 9ÿ-ÿWhat is P(X < 4)? ,c;tÿ-ÿtcqi,)-ÿ- =

What is P(3 < X < 6)? c ÿ', f ,)--Y [' qc t-ÿ ÿ[ÿ:

What isP(3<X<6)? . J'ÿ-ÿ,ÿ - , (,,bÿ

q

. Let Y denote the number of broken eggs in a randomly selected carton of one dozenSuppose that the probability distribution of Y"store brand" eggs at a certain market.

is as follows:Y 0 1 2 3 4

.

P(Y) .65 .20 .10 104

Interpret P(X 1) .20. "ÿiÿ ÿqÿ,l)ÿ ('(t J#ÿ,,t [t¢,ÿ ,ÿ2 [ Jtÿ-¢'," -ÿ'ÿ] /ÿ' ÿ'ÿ (t,,.'f'ÿz),ÿ

• Calculate the probability that the carton contains at most two broken eggs.. ÿ.ÿ-P; j,:r, 1ÿ % ÿ"

• Calculate the probability that the carton contains fewer than two broken eggs..ÿ;ÿ;r")ÿ =' 'ÿi-

• Calculate the probability that the carton contains exactly ten unbroken eggs.

• Calculate the probability that the carton has at least ten eggs that are unbroken.

Given the following density function represents a continuous random variable Z.

1

0

k

i 2

Describe the process to calculate the two probabilities. What can we say abpctLtheirprobabilitiesÿ "/7,ÿ [,tÿO,t 6ÿ /l )ÿ" a/ÿ .ÿ'ÿ.,,ÿi ÿ1" ÿ7ÿtt ÿ ÿx. q"vÿ., ÿ;;ÿ /ÿ-ÿ

What is P(Z> 1) ÿ,ÿ -,-4 ÿ ¢,Jxÿ/ÿqhat is P(Z _> 1)

8. Below is a distribution for number of cars in a household. X = # of cars in a household

X 0 1 2 3 4 5P(X) .o9 .36 .35 .la .05 .o2

Determine the expected value, variancÿandard deÿÿT-'ÿ.,ÿ- i.ilÿ?ÿ"

, The Lewis family has 3 kids. Every Sunday, Susan, the mom, buys either I or 2 gallonsof milk. Below is the probability distribution for the number of gallons of milk boughtat the store on Sunday.

X I 1 2 Find the expected value and variance.

Over Christmas, grandparents come and visit. Susan will have to buy and extra gallon of

milk. Complete the probability distribution for the number of gallons of milk bought.

Find the expected value and variance.

i;,ÿ,al: (J-;ÿ.ÿ)ÿ,J + (,ÿ--zÿÿ',ÿ: , it,

Next door to the Lewis family is the Onweagba family. They have 3 kids also.Unfortunately, the parents are sick, so Susan has agreed to do their shopping thisSunday. She will need to by double the amount of milk. Complete the probabilitydistribution for the number of gallons of milk bought.

2X ÿ t.4P(X) .2 .8 Find the expected value and variance.

Kÿ- -P' .9ÿ'Lf ; ÿ ÿ- ÿ.ÿ

¢ÿ, ÿ.,ÿ1 --. 0-3< ÿ-y,, j,ÿ (q-ÿ,ÿ,)'ÿ,, fÿ ÿ,'ÿ

ZÿJ':

L c..-z,ÿ,ÿ',ÿ' r

+(x.).qT-.tÿ

10. Now that the new models are here, a car dealership has lowered prices on last year's

models. An aggressive salesperson estimates the following probability distribution ofX, the number of cars that she'll sell next week. ÿ€ÿ ÿ 0 ÿ ÿ'ÿ t i,,ÿt)', tCt jÿ-,.ÿ5tff',ÿ

X 0 1 2 3 4P(X) .05 .15 .35 .25 .20 .Lÿ'ÿ. ÿ-- -2. H'

a. Determine the expected value, variance and standard deviation (from homework).

b. Suppose that this salesperson earns a $200 commission for each car sold. Whatare her expected wages for next week? What is the standard deviation of thosewages? "fl,,tiÿ.t : ÿ1,.ÿr,(,ÿoÿ)---_ ÿ) 4"ÿ',ÿx = ÿk/"ÿ, I-ÿL/" " ÿ. ,ÿ.ÿ, 7i

Suppose the employee's complain to the owner about their pay. They want some

type of guaranteed salary and more commission. The owner relents and decides

they get $150 each plus $250 commission for each car sold. What are herexpected wages for next week? What is the standarÿ those wages?

11. Suppose X and Y are random variables with ÿx=35, ÿrx=8,/ÿr=72, %=4. Calculate the

following:/-'ÿx =),(tÿ- 7L'

/-Ao-x = It.' " ,1ÿÿ'- ".J-'ÿ"

rex-, =,t(ÿ-)_I : IJ'l

O-2X : ,

<ÿ,,-, ='-tÿ, CW ,.(cÿ /.,ÿoo-ÿ ÿt..< -J(,D

S-ÿ.-

2<ÿ_8= It,->

<,2oo 2,

12. A club sells raffle tickets for $5 each. There are 10 prizes of $25 and one priÿe of$100. If 200 tickets are sold, determine the probability distribution. What are yourexpected winnings per ticket? Have you paid too much for the ticket (duh)? Explain.

Combining and Transforming Random Variables

13. Suppose X and Y are random variables with px=35, Gx =8,/ÿy=72, ÿ =4. Given that X

and Y are independent variables, calculate the following:

14. For a given high school basketball team, the number of baskets (X) for the leading

scorer is E(X) = 8.3 with G(X) = 1.25 and the number of baskets (Y) for the second

leading scorer is E(Y) = 6.6 with ÿ(Y) = 2.31.

a° Together, how many baskets would we expect each game? E'L'ÿ/= ÿ ?r ÿ., ÿ'

=lq1ÿb. What is the standard deviation of this number? -ÿ

c. How many more baskets would we expect the leading scorer to have?

d. What is the standard deviation of this number?

e. Hypothetically, let's say the leading scorer shoots only 3-point baskets and the

second leading scorer shoots 2-point baskets, how many points would we expect

each game from the two players.ÿ3ÿ.2y, = ÿ(ÿfÿ ÿJ.)'t ÿf.;,b) -- ÿ,

f. What is the standard deviation of this number?o--ÿ ÿ :ÿSiÿ =ÿ,(-iSÿI.ÿ7¢1,

g. How many more points would we expect,the leading scorer to have?

h. What is the standard deviation of this number?

- g, t

15. The following are the distributions for the number of goals scored per soccer game for

the three top forwards from Vareebull High School in Random, Texas.

XavierX 0 1

P(X) .10 .55

Yousuf2 Y [ 0 1 2 Z.35 P(Y)I .20 .75 .05 P(Z)

Zouÿ.

0 1 2 3.80 .10 .05 .05

a) Calculate the mean, variance and standard deviation for the number of goals thatXavier scores, fÿ -- dÿ'. I ÷ ( ", 9ÿ-t A ", $ÿ,-_- iÿ-/v,-

= /g'-taÿ?-, ÿ ÿ- (t-ÿ,z,la.,ÿ',-(O-/,.ÿr)ÿ. ÿs- = . ÿ7$" ÿ, -- .ÿdÿq¢q?7'tq

b) Calculate the mean, variance and standard deviation for the number of goals that

Yousuf scores. /<'tr - O' ,ÿ t-[., 7ÿ t.d:.(,T ÿ- . yS-

,;- -

16. Suppose X and Y are random variables with px=100, or, =10, pr=72, ar =18.

X and Y are independent variables, calculate the folloÿ.ÿ' ÿ _- 5Oq

.-- ax+y l(.ÿeV ]2q z-

Given that

O'x_Y

2m

17. Chen, Hope, Marco and Panchali love to visit the local bakery that specializes in muffins.Listed below are mean and standard deviation of the number of muffins each buys per

visit.

Chen:5_.1. Iÿ )-I

/ÿ = 2.3 muffins, ÿ = 1.1 muffins Hope:/ÿ = 3.9 muffins, ÿ = 2.2 muffins

Marco:/ÿ = 5.2 muffins, ff = 0.7 muffins=-ÿ -qq

Panchali:/1 = 4.2 muffins, ÿ7 = 1.4 muffins

a) If all four friends visit the bakery, what is the expected number of muffinspurchased? The standard deviation?/ÿ,fÿrÿ - ÿ, >ÿ ÿ-ÿ, ÿ /ÿ,ÿ tq, J -/ÿ

t-ÿ .wl'ÿtr' - . .---, - ÿ ÿ'.ÿr &r . -

b) How many more muffins wou d we expect ÿarco to purchase than Chen? Thestandard deviation? ,/'(,ÿt-c -" ÿ,J--ÿ, ÿ =- oÿ, 'i

c) How many more muffins would we expect the males to buy. than the females?The standard deviation? ÿLc40-ÿ ÿ/ÿ'I - (.],ÿr ÿ-ÿ) -(J,qÿ,d) '=" -. ÿ

"ÿ'ÿ-,ÿ.ÿ ÿÿ_-(I,ÿ,ÿÿ.ÿ,',ÿ, ¢-s.- .ÿt.,; .ÿ,)_ÿ - ÿ,qlÿq"iS-ÿ;ÿl?-

d) Hope promised to buy 4 extra muffins for her colleagues. How many muffinswould we expect her to purchase? The standard deviation?

e) The two ladies love going Thursday mornings because all muffins are $2.50 each.How much would we expect them to spend? The sÿandard deviation? g- _-%ÿ. S"

f) Marco buys the $4 mega muffins and Panchali buys the regular $3.25 muffins.How much higher will Marco's cost than Panchali's? The standard deviation?

18. The probability distribution below represents (Y) the length of long distance calls inminutes.

Y 5 10 15 20 25P(Y) .1 .2 .3 .3 .I

a. What is the expected length of a long distance call? The variance and standarddeviation? ,¢1/,: ÿ'-,It-lc;',2.€:lÿ",/.ÿ .>u', ?ÿ-_,1ÿ-: I - 15-,5- lÿ'ÿ = ÿ.,)ÿ--

b. Suÿ that there is an initial connection charge of $.60 and a further charge9ÿ"for each minute. What is the expected cost of a long distance call?

if'What is the standard deviation of the cost of the call?

.tÿ- , 1' . .- ÿ ,- : c ÿb-3ÿtÿ

_ @ÿ]" , O , r . ÿ ÿ ÿ ,! -ÿ t , ' " 'ÿ c ÿ. ÿ ' t

19. Some Japanese consumers are willing to pay premium prices for Maine lobsters. Onefactor in the high cost is the mortality rate during the long-distance shipping fromMaine to Japan. Suppose the number of deaths X per crate of 10 lobsters is given bythe following probability distribution.

XP(X)

a.

0 1 2 3 4 5

b.

0.48 0.28 0.17 0.05 0.01 0.01/,-

How many--lobsters would we expect to arrive in Japan per crate? /ÿ , ,

If one restaurant wants only the lobster claws, how many claws woÿhe . j,shipping company expect to lose from 3 crates? < ÿ , o) /c/ÿ ,

20.The random variable X denotes the number of Granny Smith apples per purchase bysupermarket shoppers in the express lane. The probability distribution is given below:

2 1 2 3P(X) 0.35 0.40 0.25

a. Find the expected value, variance and standard deviation for the number ofapples. /xy -(,,3ÿ'F ,),,q ÿ}',2ÿ' = [ÿq

; = (l-t.ÿi)a,ÿ" t- (ÿ-Dq)ÿ..q t( 3 /tcZ)J- ÿ-- $-q 4Uÿx 7ÿ-,ÿ2 qs-vqÿ

/-1

('j" =LL.;I

b. If the apples are 50¢ each, what is the expected cost? The standard deviation

',7 . ,ÿ i ." ' ÿ"

c. A shopper in the express lane is expected to spend 72¢ on bananas with a

standard deviation of 10¢. How much more would we expect the shopper to

spend on apples than bananas? The standard deviation?

--- ..

tddÿ,+.4j --