THE EGS5 CODE SYSTEM

that the reader is already acquainted with the elements of probability theory.
The primary entities of interest will be random variables which take values in certain subsets
of their range with specified probabilities. We shall denote random variables by putting a “hat”
( ˆ ) above them (e.g., ˆx). If E is a logical expression involving some random variables, then we
shall write P r{E} for the probability that E is true. We will call F the distribution function (or
cumulative distribution function) of ˆx if
When F (x) is differentiable, then
is the density function (or probability density of ˆx and
F (x) = P r{ ˆx < x} . (2.1)
f(x) = F ′ (x) (2.2)
P r { a < ˆx < b} =
In this case ˆx is called a continuous random variable.
∫ b
a
f(x)dx . (2.3)
In the other case that is commonly of interest, ˆx takes on discrete values x i with probabilities
p i and
F (x) = ∑
p i . (2.4)
We call P the probability function of ˆx if
x i ≤x
P (x) = p i if x = x i
= 0 if x equals none of the x i . (2.5)
Such a random variable is called a discrete random variable.
by
When we have several random variables x i (i = 1, n), we define a joint distribution function F
F (x 1 , . . . , x n ) = P r{ ˆx 1 ≤ x 1 & . . . & ˆx n ≤ x n } . (2.6)
The set of random variables is called independent if
P r{ ˆx 1 ≤ x 1 & . . . & ˆx n ≤ x n } =
n∏
i=1
P r{ ˆx 1 < x i } . (2.7)
If F is differentiable in each variable, then we have a joint density function given by
f(x 1 , . . . , x n ) = ∂ n F (x 1 , . . . , x n )/∂x 1 , . . . , ∂x n . (2.8)
Then, if A is some subset of R n (n-dimensional Euclidean Space),
∫
P r{ ˆ¯x ∈ A } = f(x)d n x . (2.9)
A
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