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Unformatted text preview: STAT 116 STANFORD UNIVERSITY
Department of Statistics FINAL EXAMINATION
STATISTICS 116 Instructor: Professor Anthony D’Aristotile August 17, 2007
3:30 pm  6:30 pm IMPORTANT: Please, show all your work and justify your assertions. You may
leave your answers in indicated form. There are 200 total points. STUDENT Name:
STUDENT ID :
INDICATE IF SCPD STUDENT (YES OR NO): 1. (10 + 10 points) Suppose that the heights of male Purdue students are normally distributed with mean 70.5
inches and standard deviation 3 inches. (a) What is the probability that typical Purdue male student is taller than 75 inches? (b) If you randomly sample Purdue male students one—byone until you get one taller than
75 inches, what is the expected number of men you’ll need to sample? ?<2Z> 7S'37o§'> _: V(2> ___ ?(2,(.Y) P< X>7$> : \_ 7(2615); 2. (10 + 10) (a) Suppose that X is a random variable whose moment generating function is given by
Mx(t) = 1/5e‘ + 2/56“ + 2/5e8t for —00 < t < 00.
Find the probability distribution function of X. (b) Suppose that the random variables X and Y are independent and identically distributed,
and that the moment generating function of each is et2+3t for —oo < t < 00.
Find the moment generating function of Z r 2X — 31" + 4. qﬁzI‘i‘
_ “(170 ‘2
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44 (*3 3. (20) A miner is trapped in a mine containing 3 doors. The ﬁrst door leads to a tunnel that will
take him to safety after 3 hours of travel. The second door leads to a tunnel that will return
him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him
to the mine after 7 hours of travel. If we assume that the miner is at all times equally likely
to choose any one of the doors, what is the expected length of time until he reaches safety?  1' 4
“.dggcﬂmu \ M
. 0
= 2 h u 2 y’ 4M
3 \\ “ 3 I} AM 3
3 __ 4 <g4E(X))—;
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= ‘+ 3
7 IS‘
1 Z+—= —
E00“ 3"' 3 3 3
7\ €LX)=‘S' II ll (14 + 6) Suppose that balls are withdrawn one at a time and without replacement from an urn that
initially contains n balls, of which k are considered special, in such a manner that each
withdrawal is equally likely to be any of the balls that remain in the urn at the time. For
1 S i S n, let Xi = 1 if the ith ball is special and let it be 0 otherwise. (a) Prove that the variables X 1, X 2, ..., X” are exchangeable. (b) Find the probability that the ﬁrst ball is special given that the last ball is not special.
Explain your answer. ‘ I“ t.’—’~h cg“ GM‘
P(x=?l) x1: ‘) ”’ Y’=1h) : ft5=k
a any.
x ,. 3‘ MK
1% (mm m M4 —~ w M
Md“a"'m* 5. (10 + 10)
GE has 2 plants which make 75 watt bulbs. Their plant in Churubusco, Indiana, makes 60
percent of their 75 watt bulbs and Churubusco bulbs have exponential lifetimes with mean
3 years. The remaining 40 percent of their 75 watt bulbs come from Tuscaloosa, Alabama,
and Tuscaloosa bulbs have exponential lifetimes with mean 2 years. (a) Find the probability that a random GE 75 watt bulb is still burning after 5 years. (b) Given that a random GE 75 watt bulb is still burning after 5 years, what is the proba
bility that it is a Churubusco bulb? C —W'
Lewdgce 95 “aw T__7M.Jnut
5 07"
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—_ MC
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Ml
. 4 = ,g; “3+: 7:“ 6. (8 + 8 + 4)
Suppose X and Y have joint probability density function f(x,y) = 6y when :1: > 0, y > 0, and
a: + y < 1. (a) Find P(X > (b) Find the marginal probability density functions of X and Y.
(0) Are X and Y independent? 7. (20)
Let X and Y be independent exponential distributions with means E(X) = E(Y) = 1. Find P(Y22X+1).
N ‘12 a»): S 4*” [infﬁa #31 PM
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if 8. (20)
Suppose that a random sample X 1, X 2, ..., X 300 of size 300 is taken from the uniform distribution on [0, 12]. If 5300 = 3101 X j, use the Central Limit Theorem to estimate
Pa 5300 — 1800 g 15). 1
_ (2+)
 to \{i E(><)=é “"7 it
X ,,.{o~v~ F” )
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3.. ﬂ Mk goo.é : lQoe
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2 (ﬁg?) O 9. (10 + 10)
A seven judge panel will make a recommendation in favor of or against the death penalty. Each of the members of the panel will independently vote in favor of the death penalty with
probability .7 and against the death penalty with probability .3. (a) If the panel’s decision requires 4 or more votes of support, what is the probability that
the panel decides in favor of the death penalty? (b) Given that exactly 4 of the judges voted the same way in this matter, what is the prob
ability that the panel decides in favor of the death penalty? H
J10
(a
0
VA
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W 10 { (mu) . '7;— (N'ﬁ) Q1”) ‘5 ’T “(3)
(“)V) 0 ch»! 4 10. (4 + 16)
Consider X and Y with joint probability density function f(x,y) as given in Problem 6 (a) Give a geometric description of the domain of f(x,y). (b) Find the joint probability density function of U = 2X and V = 2X +2Y. (HINT: In the
anyplane, consider the triangle with vertices (0, 0) , ( 1, O) and (O, 1) and its boundary B. _ —1 —1. “21433402233
In the uv plane, ﬁnd T (B) where T  v = “x, y) = 2n: + 2y maps the cryplane into the uvplane. ...
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