MATHEMATICS 334
Spring 2014
Homework 02
1. Let U1 and U2 be independent and uniform on [0, 1]. Find and sketch the density function of
S = U1 + U2 by variable transformation.
f (s, u2 ) = f (s u2 , u2 ) 1 = 1, 0 s 2, 0 u1 = s u2 1
(1) 0 s 1, 0 u2 s
s
f (s
MATHEMATICS 334
Spring 2014
Solution key for Practice Test 01
1. (20%)
(a.) Let X be an exponentially distributed random variable with mean
eX is uniformly distributed in [0, 1].
X Exp(), f (x) = ex , x > 0; P(X t) = et
Method01:
=
Show that Y =
P(eX y)
=
MATHEMATICS 334
Spring 2014
Practice Test 01
1. (20%)
(a.) Let X be an exponentially distributed random variable with mean
eX is uniformly distributed in [0, 1].
1
.
Show that Y =
(b.) Let X1 , X2 , . . . , Xn be an exponentially distributed random sample
Suciency
Suciency
Outline
1 Sucient Statistics
Math334 Spring 2014
Suciency
2
EM algorithm
Ping-Shi Wu
Department of Mathematics, Lehigh University
April 9, 2014
Ping-Shi Wu
Suciency
Math334 Spring 2014
Su. Statistic and Partition
Neyman-Fisher Factorizat
Gamma(, ): revisited
1
For k > , E(Xk ) =?
Math334 Spring 2014
Gamma(, ), t , F1 ,2
Ping-Shi Wu
Department of Mathematics, Lehigh University
February 26, 2014
Ping-Shi Wu
Independence:revisited
1
3
Math334 Spring 2014
Example (Remark: 2.6.1)
X1 , X2 , . .
Random Variable: X
A function mapping from to R.
1
Math334 Spring 2014
Random Variable
Ping-Shi Wu
If is countable (possibly innitly), X is discrete, e.g.
Bernoulli, Binomial, Poisson, Geometric, Hypergeometric,
Negative Binomial, Discrete Uniform, .etc.
MATHEMATICS 231
Spring 2014
Homework 01-Soln
1. a. 1.7.22
1
f (x) = , x .
2
2
1
x = arctan y, dx = 1+y2 , < y < .
dy
1 1
g(y) = 1+y2 , < y < .
b. 1.8.10
The expectation of X doesnt exist because of
2
a x
2
E(|X|) = lim 0 1+x2 dy = lim ln(1+x ) |a =
0
a
a