Reading: Ch. 1, 2 and 4 of Casella and Berger.
Goals of Today's Lecture:
Today's notes
So far: defined probability space, real and vector valued random variables cdfs in R1 and Rp, discrete densities and densities for random variables with absolutely continuous distributions.
Started distribution theory: for Y=g(X) with X and Y
each real valued
Method 2: Change of variables.
Assume g is one to one.
I do: g is increasing and differentiable.
Interpretation of density (based on density = ):
Remark: For g decreasing
but Then
the interval
is really
so
that
.
In both cases this amounts to the formula
Example:
or
Let
or
.
Solve
:
or
g-1(y) = ey. Then
and
Hence
Simplest multivariate problem:
,
Y=X1 (or in general any Xj).
is the marginal density of
and fXthe joint density of X but
they are both just densities.
``Marginal'' just to
distinguish from the joint density of X.
Example The function
General problem has
with
.
Case 1: q>p. Y won't have density for ``smooth'' g. Y will have a singular or discrete distribution. Problem rarely of real interest. (But, e.g., residuals have singular distribution.)
Case 2: q=p. We use a change of variables formula which generalizes the one derived above for the case p=q=1. (See below.)
Case 3: q < p.
Pad out Y-add on p-q more variables (carefully chosen)
say
.
Find functions
.
Define for
,
and
Choose gi so that we can use change of variables on
to compute fZ. Find fYby integration:
Suppose
with
having density fX.
Assume g is a one to one (``injective") map, that is,
g(x1) = g(x2) if and only if x1 = x2.
Find fY as follows:
Step 1: Solve for x in terms of y: x=g-1(y).
Step 2: Use basic equation:
Example: The density
Solve for x in terms of y:
Next: marginal densities of Y1, Y2?
Factor fY as
fY(y1,y2) = h1(y1)h2(y2) where
Then
Note: We show below factorization of density is equivalent to independence.
So far density of X specified explicitly. Often modelling leads to a specification in terms of marginal and conditional distributions.
Def'n: Events A and B are independent if
Def'n: Ai,
are independent if
Example: p=3
Def'n: X and Y are
independent if
Def'n:
are
independent if
Theorem