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Probability Distributions

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces the concept in probability distributions, such as probability mass function(pmf), cumulative distribution function(cdf) and probability density function(pdf).

The distribution PXPX of a random variable XX is simply a probability measure which assigns probabilities to events on the real line. The distribution PXPX answers questions of the form:

What is the probability that XX lies in some subset FF of the real line?

In practice we summarize PXPX by its Probability Mass Function - pmf (for discrete variables only), Probability Density Function - pdf (mainly for continuous variables), or Cumulative Distribution Function - cdf (for either discrete or continuous variables).

Probability Mass Function (pmf)

Suppose the discrete random variable XX can take a set of MM real values x1 … xM x1 … xM , then the pmf is defined as:

pX ⁢ xi =PrX= xi = PX ⁢ xi pX xi X xi PX xi
(1)
where ∑i=1M pX ⁢ xi =1 i 1 M pX xi 1 . e.g. For a normal 6-sided die, M=6 M 6 and pX ⁢ xi =16 pX xi 1 6 . For a pair of dice being thrown, M=11 M 11 and the pmf is as shown in (a) of Figure 1.

Figure 1: Examples of pmfs, cdfs and pdfs: (a) to (c) for a discrete process, the sum of two dice; (d) and (e) for a continuous process with a normal or Gaussian distribution, whose mean = 2 and variance = 3.
Figure 1 (figure1.png)

Cumulative Distribution Function (cdf)

The cdf can describe discrete, continuous or mixed distributions of XX and is defined as:

FX ⁢x=PrX≤x= PX ⁢ −∞ x FX x X x PX x
(2)
For discrete XX:
FX ⁢x=∑i pX ⁢ xi xi ≤x FX x i pX xi xi x
(3)
giving step-like cdfs as in the example of (b) of Figure 1.

Properties follow directly from the Axioms of Probability:

  1. 0≤ FX ⁢x≤1 0 FX x 1
  2. FX ⁢−∞=0 FX 0 , FX ⁢∞=1 FX 1
  3. FX ⁢x FX x is non-decreasing as xx increases
  4. Pr x1 <X≤ x2 = FX ⁢ x2 − FX ⁢ x1 x1 X x2 FX x2 FX x1
  5. PrX>x=1− FX ⁢x X x 1 FX x
where there is no ambiguity we will often drop the subscript XX and refer to the cdf as F⁢x F x .

Probability Density Function (pdf)

The pdf of XX is defined as the derivative of the cdf:

fX ⁢x=ddx FX ⁢x fX x x FX x
(4)
The pdf can also be interpreted in derivative form as δ⁢x→0 δ x 0 :
fX ⁢x⁢δ⁢x=Prx<X≤x+δ⁢x= FX ⁢x+δ⁢x− FX ⁢x fX x δ x x X x δ x FX x δ x FX x
(5)
For a discrete random variable with pmf given by pX ⁢ xi pX xi :
fX ⁢x=∑i=1M pX ⁢ xi ⁢δ⁢x− xi fX x i 1 M pX xi δ x xi
(6)
An example of the pdf of the 2-dice discrete random process is shown in (c) of Figure 1. (Strictly the delta functions should extend vertically to infinity, but we show them only reaching the values of their areas, pX ⁢ xi pX xi .)

The pdf and cdf of a continuous distribution (in this case the normal or Gaussian distribution) are shown in (d) and (e) of Figure 1.

Note:

The cdf is the integral of the pdf and should always go from zero to unity for a valid probability distribution.

Properties of pdfs:

  1. fX ⁢x≥0 fX x 0
  2. ∫−∞∞ fX ⁢xdx=1 x fX x 1
  3. FX ⁢x=∫−∞x fX ⁢αdα FX x α x fX α
  4. Pr x1 <X≤ x2 =∫ x1 x2 fX ⁢αdα x1 X x2 α x1 x2 fX α
As for the cdf, we will often drop the subscript XX and refer simply to f⁢x f x when no confusion can arise.

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