Denition: Let (X1 , X2 , . . . , Xn )T be a random vector. Then the random variables in the vector are independent if, for all subsets B1 , B2 , . . . Bn B
PX1 ,.,Xn (B1 Bn ) = PX1 (B1 ) PXn (Bn )
Ie. the events X1 B1 , . . . Xn Bn are
3. Random Vectors and Joint Distributions.
3.1 Random Vectors.
So far weve only considered one random variable at a time.
Now we are going to consider more than one (but at most a nite collection of) random variables.
Suppose X1 , X2 , . . . , Xn are n
2.5 The Normal Distribution
There is one absolutely continuous distribution that must be singled out as possibly the most important
and/or pervasive in all of probability theory: the normal distribution.
Denition: Any absolutely continuous random variab
2. Random Variables and Distributions.
2.1 Random Variables
The experiment space (S, S , P ) can be very abstract. Nothing rules out the possibility that S is a set whose
elements are H s and T s or even cat, dog and sh for that matter.
Many times a sci
Denition: The experiment: is the procedure or phenomenon that generates a random outcome.
1. Probability Models and More.
Flipping a coin and recording the outcome.
1.1 Probability Models.
Waiting at the bus stop and recording the waiting ti
3.5 Multivariate Normal Distribution
Dention: A random vector (X1 , X2 , . . . , Xn )T is said to have the multivariate normal distribution with
= ( , . . . , )T
if it has joint probability density function given by
fX1 ,.,Xn (x1 ,
VOLUMES AS ITERATED
A continuous function f (x, y) of two variables can be integrated over a plane re-
gion R in much the same way that a continuous function of one variable can be
integrated over an interval. The result is
3.6 Conditional Distributions
Recall that for random vector (X1 , X2 , . . . , Xn )T the joint cumulative distribution function FX1 ,.,Xn completely describes the joint distribution of the variables and is given by:
FX1 ,.,Xn (x1 , . . . , xn ) = PX1 ,.,
3.8 Multidimensional Change of Variable Formula (n = 2)
When X is a random variable with distribution PX (discrete of absolutely continuous) weve already considered
the problem of nding the distribution, PY , of Y = h(X ), where h : R R.
Now we turn our