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Examples of calculations using these functions are shown next:
Continuous probability distributions
The probability distribution for a continuous random variable, X, is
characterized by a function f(x) known as the probability density function (pdf).
The pdf has the following properties: f(x) > 0, for all x, and
. 1 ) ( =
∫
∞ +
∞ −
dx x f
Probabilities are calculated using the cumulative distribution function (cdf), F(x),
defined by
∫
∞ −
= = <
x
d f x F x X P
ξ ξ ) ( ) ( ] [
, where P[X<x] stands for “the
probability that the random variable X is less than the value x”.
In this section we describe several continuous probability distributions
including the gamma, exponential, beta, and Weibull distributions. These
distributions are described in any statistics textbook. Some of these
distributions make use of a the Gamma function defined earlier, which is
calculated in the calculator by using the factorial function as
Γ
(x) = (x-1)!, for
any real number x.
The gamma distribution
The probability distribution function (pdf) for the gamma distribution is given
by
; 0 , 0 , 0 ), exp(
) (
1
) (
1
> > > − ⋅ ⋅
Γ
=
−
β α
β α β
α
α
x for
x
x x f
P X x F x f d
x
[ ] ( ) ( ) . < = =
−∞
∫
ξ ξ