Introduction to Probability and Statistical Theory
BIOS 341

Spring 2011
Exam 1, Biostatistics 341
Please show all your work and perform all calculations to Whatever degree of exactness you
are able. This test is closed book and no calculators are allowed.
1. (20) Five fair dice (6sided with equal probability for each side)
BIOS 342
Homework 3 Key
February 28, 2013
2
2
1. Let (X, Y ) Bivariate Normal, with E (X ) = x , E (Y ) = y , V ar(Y ) = Y , V ar(X ) = Y , and
Corr(X, Y ) = . An unbiased estimator of Y is Y itself.
a. Show that Y, as an estimator, can be improved by con
BIOS 342
Homework 4 Key
February 6, 2013
1. Suppose we use a gamma distribution as our working model to learn about the expected value of i.i.d.
random variables X1 , ., Xn when we know the data really come from some unknown distribution Xi
g (X ) that i
BIOS 342
Homework 3 Key
February 3, 2013
1. Let X U ( 1, + 1) with prior distribution P () 1/ where > 0:
a. Show that P (x) = c/, (x  1, x + 1), where 1/c = log[(x+1)/(x1)], x>1.
P (X )P ()
P (X )P ()
P (X )
1
1
1
1cfw_>0 = 1cfw_>0 1cfw_>0
=
( + 1)
BIOS342  Homework 1
January 10, 2013
Let Z be the state of disease with Z = 1 for diseased and 0 otherwise. Let the prevalence of disease
be = P(Z = 1). Let X be the symptoms presented by that individual. When Z = 1 we have that
X f(X) and when Z = 0 we
1. Let X1, . . . ,Xn N t(a:) and suppose we have two hypotheses: Hf : X N f(a:) and Hg : X N Show
that the likelihood ratio for H f versus Hg converges to support the distribution that is closest to the true
density, t(:r), in terms of KullbackLeibler d