Reading for Today's Lecture:
Goals of Today's Lecture:
Review of end of last time
Proof: Reduces to
and
.
Step 1: Define
Notice ftn of y1 times
ftn of
.
Thus
is independent of
.
Since sZ2 is a function of
we see that
and
sZ2 are independent (remember that
).
Also: density of Y1 is a multiple of the function of y1 in
the factorization above. But this is the standard normal
density so
.
First 2 parts of theorem done. Third part is homework exercise
but I will outline the derivation of the
density.
Suppose
are independent N(0,1). Define
distribution to be that of
.
Define angles
by
(Spherical co-ordinates in n dimensions. The
values run
from 0 to
except for the last
whose values run from 0 to
.)
Derivative formulas:
Fourth part of theorem is consequence of
first 3 parts of the theorem and definition of distribution:
if it has same distribution
as
Derive density of T in this definition:
Differentiate wrt t by
differentiating inner integral:
Two elementary definitions of expected values:
Def'n If X has density f then
Def'n: If X has discrete density f then
If Y=g(X) for a smooth g
by the change of variables formula for integration. This is good
because otherwise we might have two different values for E(eX).
In general, there are random variables which are neither absolutely continuous nor discrete. Here's how probabilists define E in general.
Def'n: RV X is simple if we can write
Def'n: For a simple rv X define
For positive random variables which are not simple we extend our definition by approximation:
Def'n: If
then
Def'n: We call X integrable if
Facts: E is a linear, monotone, positive operator:
Major technical theorems:
Monotone Convergence: If
and
(which has to exist) then
Dominated Convergence: If
and
rv X such that
(technical
details of this convergence later in the course) and
a random variable Y such that
with
then
Fatou's Lemma: If
then
Theorem: With this definition of E if X has density
f(x) (even in Rp say) and Y=g(X) then
Works, e.g., even if X has a density but Y doesn't.
Def'n: The
moment (about the origin) of a real
rv X is
(provided it exists).
We generally use
for E(X). The
central moment is
Def'n: For an Rp valued random vector X we define
to be the vector whose
entry is E(Xi)(provided all entries exist).
Def'n: The (
)
variance covariance matrix of X is
Moments and probabilities of rare events are closely connected as will
be seen in a number of important probability theorems. Here is one
version of Markov's inequality (one case is Chebyshev's inequality):
The intuition is that if moments are small then large deviations from
average are unlikely.
Example moments: If Z is standard normal then
and (integrating by parts)
so that
If now
,
that is,
,
then
and
Theorem: If
are independent and each Xi is
integrable then
is integrable and
Proof: Suppose each Xi is simple:
Def'n: The moment generating function of a real valued X is
Def'n: The moment generating function of
is
Formal connection to moments:
Sometimes can find power series expansion of
MX and read off the moments of X from the coefficients of
tk/k!.
Theorem: If M is finite for all
for some
then
The proof, and many other facts about mgfs, rely on techniques of complex variables.