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
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