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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
Probabilities are calculated using the cumulative distribution function (cdf), F(x),
defined by , 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
The corresponding (cumulative) distribution function (cdf) would be given by an
integral that has no closed-form solution.
The exponential distribution
The exponential distribution is the gamma distribution with a = 1. Its pdf is
given by
PX x Fx f d
x
[]() ().
<= =
−∞
∫
ξξ
.1)( =
∫
∞+
∞−
dxxf
∫
∞−
==<
x
dfxFxXP
ξξ
)()(][
;0,0,0),exp(
)(
1
)(
1
>>>−⋅⋅
Γ
=
−
βα
β
αβ
α
α
xfor
x
xxf