Download Probability Spaces

1 - Probability Spaces and Random Variables
A random variable X has a gamma distribution with parameters m > 0 and  > 0 if its
probability density function has the form f(t) = f(t; m,) =
tm-1e-t/
for t > 0. If X1, …, Xn
 m (m)
are independent gamma random variables with Xj having parameters mj and j and
Y = X1 +  + Xn is their sum, then Y has density function
A(t) = f(t;m1,1) * f(t;m2,2) *  * f(t;mn,n). Here * denotes convolution. One can not
express A(t) in terms of elementary functions except if the mj are all integers or if the j
are all equal. These notes contain the background for approximations to A(t) along with
error bounds for these approximations. These approximations are useful for obtaining
both qualitative and quantitative information about A(t).
In this first chapter we begin with a short summary of the basic properties of probability
and random variables that will be used in the later chapters. For more information on
probability theory, see Loève [10].
1.1
Probability Spaces
The starting point of probability are probability spaces.
Definition 1. A probability space is a triple (, F, P) where

= a set, called the sample space
F
= a -algebra of subsets of 
P
= a measure on (, F) such that P() = 1, called a probability
measure
In the following

= a generic element of , called an outcome
E
= a generic element of F, called an event
P(E)
= probability of the event E occuring
1.1 - 1
Example 1. Let  = R, F = Borel subsets of R, and f(t) be a non-negative, integrable

real-valued function on R with 
 f(t) dt = 1. Let P(E) = 
 f(t) dt for E  F. Then
(, F, P) is a probability space.
E
-
In the remainder of these notes (, F, P) will denote a probability space, which will be
fixed unless stated otherwise.
Definition 2. Let E and F be events with P(F)  0. Then
P(E | F)
=
P(EF)
P(F)
= conditional probablility of E occuring given that F has
occurred
Definition 3. Let E and F be events. Then
E and F are independent  P(EF) = P(E)P(F)
Note that if P(F)  0 then E and F are independent  P(E | F) = P(E)
1.1 - 2