Homework Set #14 Solyutions
Stat 420 Fall 2011
1. #6.74 (a) The CDF for Y(n) is found as follows. For y (0, ),
P (Y(n) y ) = P (all Yi y ) = n=1 P (Yi n) = F (y )n = (y/)n .
i
(b) We integrate to get the pdf for Y(n) :
ny n1 /n on (0, ), and 0 elsewhere.
Homework Set #1 solutions
Stat 420 Fall 2011
1. (a) The randomly chosen student both went to a party last Saturday night and
enjoys studying statistics.
(b) The event B A is that the student went to a party and does not enjoy
studying statistics. P (B A)
Homework Set #2 solutions
Stat 420 Fall 2011
1. (Make Venn diagram)
(a) 0.011
(b) 0.0573
(c) 0.00282
(d) Not independent: a smoker is more likely to have emphysema than
nonsmoker.
2. 5/6 (make tree diagram)
3. (a) 0.10
(b) Not independent. If M represents
Homework Set #3 solutions
Stat 420 Fall 2011
1. (a) 40%; (b) 40; (c) 0.72; (d) .05
2. The probability that Sam wins exactly 3 times is .134. The probability
that he wins at least twice is 0.465. We make the usual binomial
assumptions: the trials are indep
Homework Set #4 solutions
Stat 420 Fall 2010
1. (a) yes; (b) no; not independent (also p not the same); (c) no; p not the same; (d) yes;
(e) no; maybe at church the proportion of people in support is larger.
2. Let Y be the number of times Sam plays. Then
Homework Set #5 solutions
Stat 420 Fall 2011
1. Solve using tree diagram or Bayes theorem. Answer is .711
2. Solve using tree diagram or Bayes theorem. Answer is .346
3. The test size is = .050, and the power is .61.
4. #3.132: Let the random variable Y =
Homework Set #6 Solutions
Stat 420 Fall 2011
1. (a) .0081, (b) .6555, (c) .7257, (d) 3.0 and 3.5
2. #4.2. (a) p(k ) = .2, for k = 1, 2, 3, 4, 5.
(b)
F (y ) =
0
.2
.4
.6
.8
1
for y < 1
for 1 y < 2
for 2 y < 3
for 3 y < 4
for 4 y < 5
for y > 5
(c) P (Y < 3)
Homework Set #7 solutions
Stat 420 Fall 2011
1. (I meant to say standard deviation 20 and 10, not variance 20 and 10,
but we can still do the problem.) There are a lot of dierent but correct
answers to (a), (b), and (c). The answers must demonstrate knowl
Homework Set #8 due Friday, October 21
Stat 420 Fall 2011
1. Using R or another similar package, simulate the distribution of Y =
Y1 + Y2 , where Y1 and Y2 are independent uniform random variables
on (0, 1). Make a histogram of at least 10,000 simulated v
Homework Set #11 solutions
Stat 420 Fall 2011
1. #5.136 a) E (Y ) = E (E (Y |) = E () = 1.
(b) V (Y ) = E [V (Y |)] + V (E (Y |)] = E () + V () = 2
(c) No, that is more than ve standard deviations away from the mean.
2. #5.138 (a) Using the formula, E (Y
Homework Set #12 solutions
Stat 420 Fall 2011
1. #6.8 (a) Let G be the cdf for the Beta(, ) density, and let g be its density
function. Then for u (0, 1),
F (u) = P (U u) = P (1 Y u) = P (Y > 1 u) = 1 G (1 u).
The density for U is
f (u) = g (1 u) =
(1 u)
Homework Set #13 due Friday, December 2
Stat 420 Fall 2011
1. The moment generating function of an exponential random variable
with mean = 2 is
mY (t) = (1 2t)1 .
Then for U = Y1 + Y2 ,
mU (t) = E (eU t ) = E (et(Y1 +Y2 ) ) = E (etY1 )E (etY2 ) = mY (t)2