Download Lecture 25

AMS 311
May 2, 2000
Two Problem Quiz next Thursday, May 4: univariate transformations, bivariate
transformations. Quiz returned in exam week office hours.
Final Examination: Thursday, May 11, 8:30-11:30, Old Chemistry 144.
Exam week office hours:
Professor Finch, Math 1-112: Friday, May 5: 2:30-4; Monday, May 8, 10-12;
Tuesday, May 9, 10-12; Wednesday, May 10, 10-12.
Mr. Wang, Math 3-129: Monday, May 8, 7-9 pm;
Tuesday, May 9, 7-9 pm; Wednesday, May 10, 7-9 pm.
Another problem using two variable transformations
Let X and Y be independent, identically distributed random variables that are exponential
with mean 1. Find the joint pdf of X-Y and X+Y.
Conditioning on random variables.
Recall that E ( X |Y )   (Y ) is a random variable (called the regression function).
There are two fundamental identities:
E ( E ( X |Y ))  E ( X ). This is deceptively easily stated. Make sure that you understand the
probability measure governing each expectation.
Similarly, var( X |Y ) is also a random variable (it is also a function of Y).
The second fundamental identity is
var( X )  E (var( X |Y ))  var( E ( X |Y )).
The second fundamental identity is reflected in the basic identity of the statistical analysis
of linear models: Total sum of squares=sum of squares due to model and sum of squares
due to error.
The most common example of correlated random variables is that of the bivariate normal
distribution.
Central Limit Theorems
The most basic applications are concerned with sums and averages of random samples.
Remember that a random sample of size n from the random variable X is defined to be a
set of n independently and identically distributed random variables, each with the same
marginal distribution as X .
As you learned in AMS 310, the two basic random variables we are concerned with are
n
Sn 
n

i 1
S
X i and X n  n 
n
X
i 1
n
i
.
The basic moment calculations that we studied in the last chapter give you that
E ( S n )  nE ( X ) and E ( X n )  E ( X ).
The variance calculations give you that
var( S n )  n var( X ) and var( X n )  var( X ) / n.
The central limit theorem adds the fact that the distribution of these random variables
becomes closer to normal as the number in the random sample increases.
Modern proofs use a function called the moment generating function. For those who
know complex analysis, there is a generalization of the moment generating function
called the characteristic function that is preferred (because it always exists).
The moment generating function of the random variable X is defined to be E(eXt), when it
exists. The reason for the name is that the moments of a random variable can be obtained
by success differentiation of the moment generating function.
Example Problems
Find the moment generating function for a Bernoulli trial with probability of success p.
Find the moment generating function for a binomial random variable with n trials and
probability of success p.