TEST TOPICS AND SAMPLE PROBLEMS FOR FIRST IN-CLASS TEST, Fall 2014
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I promised that the test topics and problems would be taken from
those covered in class and in the homework problems up through
the course coverage and first 3 Homework sets so far.
In brief, these are:
(i) Mixture distributions including mixed-type discrete and
continuous distribution functions.
(ii) Transformations of random variables, including: univariate
transformations (not necessarily 1-to-1) of continuous r.v.'s;
multivariate smooth and smoothly invertible transformations of
random vectors with joint densities; and the Probability Integral
Theorem.
(iii) Stochastic simulation as a problem of finding functions of
sequences of iid Uniform[0,1] random variables U_1, U_2, ...
which give random vectors with prescribed densities.
(iv) Sufficient conditions for switching the order of
differentiation with respect to parameters and integrals or of
infinite summation and integrals.
(v) Definitions and uses of (joint) moment generating functions
to establish distributional properties.
(vi) Conditional expectations, repeated conditioning, unconditional
and conditional variance formulas. Hierarchical (multistage)
definitions of distributions of random variables.
(vii) Location and scale families of densities.
(viii) Definitions and properties of multivariate-normal (possibly
singular) random vectors. Calculations with normal conditional
probability distributions.
(ix) Definitions of order statistics, and calculations with joint
and marginal and transformed densities of order statistics.
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Sample test problems:
(I) All five problems on the sample test from 10/2/01 that can be
found in the first of the three links in the "Handouts & Other Links"
part (I), at http://www.math.umd.edu/~evs/s700.F09/SampTest.pdf.
But note the correction of Problem 2 given at the end of that pdf,
and try to do only that version of Problem 2.
(II) Problems (2) and (4)(b) on the first Stat 700 In-Class Test
from Fall 2009, available as the second link in the "Handouts & Other Links"
part (I), at http://www.math.umd.edu/~evs/s700.F09/SampTest.pdf.
HINT for this problem: start by writing down the joint density of
(|X|, |Y|), using the fact that (X,Y),(-X,Y), (X,-Y) and (-X,-Y) all
have the same densities. Then do a transformation (|X|, |Y|) --> (|X|, X^2+Y^2).
(III) For simple univariate transformation problems: see problems TRAN.1
and TRAN.2 on p.3 of the handout http://www.math.umd.edu/~evs/s400/RNG1.pdf.
For simulation problems: see problems Sim.1 and Sim.2 on p.3 of the
handout http://www.math.umd.edu/~evs/s400/RNG2.pdf.
Both of these handouts are links under "Handouts & Other Links"
part (V).
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(IV) Here a few more sample problems, which are in scope for the Test but
vary in difficulty:
Sample F14.1: Suppose that X_(1), X_(2), ... , X_(n) are order statistics
from a continuously distributed sample X_1,...,X_n ~ f(x,mu,sigma) with
one of a location-scale family of densities. Prove that for each j=1,...,n,
the density of X_(j) is also of location-scale family form.
Sample F14.2: Show that the F_{m,n} type random variable is a monotonic
univariate transformed Beta(m,n) random variable. HINT: use what we proved
in class, that if U ~ Gamma(alpha,lambda) and V ~ Gamma(beta, lambda), then
U/(U+V) ~ Beta(alpha,beta).
Sample F14.3: Show that if X_1, X_2, ... X_L are a finite or infinite list
(L<= infty) of (not necessarily identically distributed) normal random
variables, then the mixture random variable with density sum_{j=1}^L c_j f_{X_j}(x)
with sum_{j=1}^L c_j=1 and all c_j > 0 is NOT normally distributed.
Sample F14.4: Suppose that A ~ Binom(1,p) is a coin-toss random variable,
with 0