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Unformatted text preview: NAME 50 L U T I” M; ’ Exam 2 Grade:
ACSC/MATH 347/447 Exam #2 100 Points Total April 15, 2010 Course Average: You must show enough work, or give sufﬁcient explanation, in each problem to clearly indicate how you obtain
your answer. No credit will be given for a problem if there is insufﬁcient work/explanation. You may not use the probability or statistical functions on a calculator. However you may leave binomial
coefﬁcients and indicated sums or products in your answers unless otherwise directed. 1. (5 points) A sample of 3 items is selected at random from a box of 20 items, of which 4 are defective.
Let Y denote the number of defective items in the sample. What is the variance of Y? Explain. 2. (10 points) You arrive at a bus stop at 10:00, knowing that the bus will arrive at some time uniformly
distributed between 10:00 and 10:30. If at 10:15 the bus has not yet arrived, what is the probability that you
will have to wait at most an additional 10 minutes? Completely evaluate your answer. Note: If Y denotes the time that you wait at the stop until the bus arrives, then Y has a U (0, 30) distribution
and the problem is asking for P(Y S 25  Y> 15). P(($'<\ré.2 )
P(Yéls') Y>/;) = P( Y 7/5?
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30,15 . / 3 ll l 3. (12 points) A fair die is rolled until a 6 appears (assume the rolls are independent).
a. What is the probability that at most three rolls are needed? (Deﬁne and use a random variable.) b. How many rolls are required so that the probability of getting a 6 is at least 0.95 ?
Completely evaluate your answer. 9—1—3— F(Y—"= 3):1’P[\/>3):l_23:i:—M1
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W Ysz) : P(‘/=1)+P(Y=2)+P(Y= 3) : ‘6'? + (a???) +(t)(%f 10.47911. £61) ((12 wdﬁ%.d%;W§Wm%gJ/LW
40W P(Yén)20.?§: '
0.?5 s PtYén) =1~P(Y>n)=1 4%) 4. (12 points) You play the following game. Three fair dice are independently rolled, and if the number 6
appears 1' times, i = l, 2, 3, then you win 1' dollars; otherwise you lose $1 (or win —1 dollar). Let Y denote your winnings in the game, so that Support(Y) = {—1, 1, 2, 3}.
a. Find the probability function of Y. b. Find the momentgenerating function of Y. 5. (12 points) Let Y be a normal random variable with mean 12 and variance 4.
Completely evaluate your answers in this problem. And draw pictures! a. Find P(ll < Y< 14).
b. Find the 90th percentile of Y; that is, ﬁnd the number 0 so that P(Y§ c) = 0.90. SW Y'v/Vmﬂ), Z = Y}? ~/W0/1), Norma/ Table (a) 3 _;' nn. ‘941
____ l(H<Y4/‘/)»Pv—i—‘<Z< x) =P(——'5_< 24.1.) 4%; = 1 ~ P(z >1)—P(2<~g), 3” 4 :> P( Z > CELIALia) :O [O 5;) 0690
p 4% of0
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W‘ﬁﬂ'zvvm Qva/\;l9')n 20—1.}?—r73§(0‘010)=/232 M c«!; ,1 = (“13724 => c : 231+ ;(i\a5>2)=/‘f.5‘6’7’ [W09 Zo=/‘;J.5>J we W C: 151+ 10‘2“?) =/‘1$:é]
m T1333/P4'5 mwom(o.7,/2,23W #363033 5. (12 points) Let Y be a normal random variable with mean 12 and variance 4. W N
Completely evaluate your answers in this problem. And draw pictures! 5 0A” a. FindP(ll <Y<14)_ fl 77.3Z b. Find the 90th percentile of Y; that is, ﬁnd the number c so that P(Y g c) = 0.90. ‘ W Viv/WWW), Z : Y}? m /\/[0/:L)‘
€99 PUI<Y< w): P,(,H_:B < Y< Lgl = N“; < 2 <4) 4%:
= Wyn—Hike) .Z 1
= Nam—N935 67 = P(z<1)~[1P(%é’ill
.__ O‘?L{_(3_ [1 —0. @6715] 20.53259 X LIL”) P(\/£C)=Oo90 :> P(Z s 6—H) :O«?O £0 20 {Mae 73>:é ”J11”
awn? wig/“2.59m Ag?) 0), 20: /\2ié>+.{%6(0,0/0):{.2Po?
W T :Lng :3 C :101+D.(/‘3?2C)=/‘/\5“6§‘
[My Zo =/~25>;we We C:/a+2{i.;é>):l%s‘6]
m 7:92/W c‘nvNorm[0.7, 51.50% 1% 563103 :3. 6. (12 points) The time required to repair a machine has an exponential distribution with mean 2 hours.
a. What is the probability that a repair time exceeds 2 hours? b. If ﬁve of these machines need repair, what is the probability that at most one of them has a repair time
that exceeds 2 hours? 7. (15 points) Let Y be a random variable with probability density function nyﬁ
i, —lsyso 1 ‘C
2
f(y)= y, 09:1 V2 eUgj‘é)
0, elsewhere.
—2 —1 'ﬁ 0 ‘3‘. 1 2 a. Represent the probability P (~ l< Y< <%) on the graph of f 0/) above and use geometry to calculate it. 2—
b. Find the mean of Y.
c. Find the distribution function, F (Y ) of Y. @P('ﬁ YeaV W%W*W%‘ﬁ
= (i)(L)+(il lléﬂél =¢+E=igﬂge
(1;) Emiﬁwwa = heady64 0% = at, + 3;; 40—week; (£3 Fly): :0 “’5de
5151: Fly)=0 # at
4‘30 ”W‘F(“*Lﬁb&=0+ﬁill=é%ﬂ$
o<&< 1‘ H5)— P/o)+§gt#=J—+[t :1: _.L @110)
bl>l1Flﬁ)=—’L ' 8. (10 points) The number of calls that arrive at a help desk during a one—hour period has a Poisson distribution
with a mean of 10. You have the morning shift and arrive precisely at 8:00 am. for your shift. a. Let N be the number of calls that arrive during the ﬁrst half—hour of your shift. Find P(N 2 3).
b. Let W be the time in hours between the arrivals of your ﬁrst call and your second call. Find P(W< 0.1). c. (10 points extra credit) Let Y be the time in hours between the start of your shift and the arrival of your
second call. Find P(Y< 0.5) . 8. (10 points) The number of calls that arrive at a help desk during a onehour period has a Poisson distribution
with a mean of 10. You have a half—hour shift at the help desk. a. Let N be the number of calls that arrive during your shift. Find P(N Z 3). b. Let W be the time in hours between the arrival of your ﬁrst call and the arrival of yoursecond call.
Find P(Ws 0.1). c. (10 points extra credit) Let Y be the time in hours between the start of your shift and the arrival of your
second call. Find P(Y< 0.5) . £52., KWWWMW ﬁlm/W) A46 4, \ WMWMV=QWJ €337 lo) H
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9. (12 points) Y has density functionf(y)={a + y ’ y } and mean E(Y) = 3. Finda and b.
0 , elsewhere 5 Hint: Use (1) the fact that f (y) is a density function and (2) the fact that E (Y) : g to get a pair of simultaneous linear equations in a and b. 05 0 2 7 4. _LZ' _. a :‘L’O’+ c 50 ~22— —z——+~$lo 1 4
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 Statistics, Probability

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