1. 9.58, 9.59, 9.80, 9.82, 9.83
2. It is known that the probability p of tossing heads on an unbalanced coin is either 1/4 or 3/4. The
coin is tossed twice and a value for Y , the number of heads, is observed.
(a) What are the possible values of Y ?
(b) F
1. We are interested in testing whether or not a coin is balanced based on the number of heads Y on 36
tosses of the coin. (H0 : p = .5 versus Ha : p 6= .5). If we use the rejection region |y 18| 4, what is
a. the value of ?
b. the value of if p = .7?
2.
1. Y is a random variable. Y (1, 1). The pdf is p(y) = ky 2 for some
constant, k.
(a) Find k.
(b) Plot the pdf.
(c) Let Z = Y . Find the pdf of Z. Plot it.
2. U is a random variable on the interval [0, 1]; p(u) = 1.
(a) V = U 2 . On what interval does V l
1. 8.39
[Sol] By = 4 degrees of freedom(the value of the parameter in 2 ), we can write
Y
2.05 ) = .90
P (.710721 X 9.48773) = .90
2Y
2Y
P (
) = .90.
9.48773
.710721
P (2.95 2
2Y
2Y
Hence the interval ( 9.48773
, .710721
) forms a 90% confidence interval
1. 8.39, 8.40, 8.58
2. Suppose that Y is normally distributed with mean 0 and unknown variance 2 . Then Y 2 / 2 has a 2
distribution with 1 df. Use the pivotal quantity Y 2 / 2 to find a
(a) 95% confidence interval for 2 .
(b) 95% upper confidence limit f
1. 8.1, 8.3, 8.4, 8.5(a,b), from the textbook
2. 8.15, 8.17(a,b) from the textbook
3. The number of breakdowns per week for a type of minicomputer is a random variable Y with a Poisson
distribution and mean . A random sample Y1 , Y2 ,., Yn of observations
1. Y is a random variable. Y (1, 1). The pdf is p(y) = ky 2 for some
constant, k.
(a) Find k.
(b) Plot the pdf.
(c) Let Z = Y . Find the pdf of Z. Plot it.
2. U is a random variable on the interval [0, 1]; p(u) = 1.
(a) V = U 2 . On what interval does V l
1. 9.39, 9.40, 9.42
2. We consider a random sample of size 3 from an exponential distribution with the parameter . let
1 = Y1 , 2 = (Y1 + Y2 )/2, 3 = (Y1 + 2Y2 )/3, and 4 = Y . Find the efficiency of 1 relative to 4 , of
2 relative to 4 , and of 3 relativ
Stat 516: Statistics II
Problem Set 2: Stat 515 Review (Chapter 6 & 7)
October 1, 2015
Name:
1. (10 points) (WMS 6.45)
2. (10 points) (WMS 6.74)
3. (10 points) (WMS 6.75)
4. (10 points) (WMS 6.88)
5. (10 points) Think about the solution of question 4 in t
1. suppose one has the set of n data points, cfw_(xi , Yi = yi ) for i = 1, . . . , n, and one wishes to fit the
model E(Yi | xi ) = 0 + 1 xi to n data points. Here, Y1 , . . . , Yn are independent r.v. If 0 and 1 are
estimators of 0 and 1 , Y = 0 + 1 x i