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<p<1, and that U, V, W, X  are any four independent, continuously 
distributed random variables with densities. Let T = (U,V)  whenever  A=1, and 
otherwise (when A=0)  T=(W,X).  Show that the components of the bivariate random 
vector T  are independent if and only if  U  and W  have the same density or 
V and X have the same density.

Sample F14.5: Find the probability mass function for U_(3) , the third order 
statistic among a sample of 5 discrete-uniform random variables U_1,...,U_5 each 
taking values {1,2,...,8} with equal probabilities 1/8.
NOTE: order statistics are allowed to be tied for samples from discrete 
probability mass functions !

Sample F14.6: Find the density of  X_(1)/X_(2)  for two independent 
Gamma(3,1) random variables.