Homework for 2/10 Due 2/19
1. [8-69] Use the factorization theorem (Theorem A in Section 8.8.1) to
conclude that T =
Xi is a sucient statistic when the Xi are an i.i.d.
sample from a geometric distribution.
The joint pmf of the random samp
History is the study of change over time. The primary task of your Country Report will be to describe how
your country changed from 1945. There may be periods of rapid change and also periods of slow
evolution - you will have to select which periods of ch
History Paper: the world since 1945
The World since 1945
History is the study of change over time. The primary task of your
Country Report will be to describe how your country changed from 1945.
There may be periods of rapid chang
the test statistic and the P-value.
The P-value of the test, assuming H0 is true, is the probability
that the test statistic would take a value as extreme or more
extreme than that actually observed.
After assessing the consequen
Homework for 1/8
1. [4-6] Let X be a continuous random variable with probability density
function f (x) = 2x, 0 x 1.
(a) Find E[X ].
(b) Find E[X 2 ].
(c) Find Var[X ].
(a) We have
E[X ] =
x 2x dx =
xf (x) dx =
(b) We have
Homework for 1/22 and 1/24
1. [6-4] Let X1 , X2 , . . . , X8 be i.i.d. normal random variables with mean
and standard deviation . Dene
S 2 /n
where X is the sample mean and S 2 is the sample variance.
(a) Find 1 such that P(|T | < 1 ) =
Homework for 1/13 Due 1/22
1. [5-23] An irregularly shaped object of unknown area A is located in the
unit square 0 x 1, 0 y 1. Consider a random point distributed
uniformly over the square; let Z = 1 if the point lies inside the object and
Homework for 1/27 Due 2/5
1. [8-13] In Example D of Section 8.4, the pdf of the population distribution
1 + x 1 x 1
f (x|) =
1 1 ,
and the method of moments estimate was found to be = 3X (where X
is the sample mean of the ran
Homework for 2/3
1. Determine the values of the following quantities:
a. t0.1,15 = 1.341
d. t0.05,40 = 1.684
g. 2.1,20 = 12.44
j. 2.99,20 = 37.57
1. A random variable whose natural logarithm follows a normal distribution
is called a lognormal random variable. In particular, if Z N (0, 1), then
X = e+Z is a lognormal random variable with parameter and . The
pdf of X is
(ln x )2
1. Let X be a continuous random variable with density function
f (x) =
1 x 1
(a) Find the mean and variance 2 of X .
(b) If S = X1 + X2 + + X60 , where X1 , . . . , X60 are i.i.d. random
variables with pdf f (x), wh
The Chi-Squared Distribution
The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known
distributions such as the normal distribution and the exponential distribution, the chi-squared
distribution is rarely used to mo