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Probability Distributions

Module by: Nick Kingsbury

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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:

pXxi=PrX=xi=PXxi pX xi X xi PX xi (1)
where ∑i=1MpXxi=1 i 1 M pX xi 1 . e.g. For a normal 6-sided die, M=6 M 6 and pXxi=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:

FXx=PrX≤x=PX-∞x FX x X x PX x (2)
For discrete XX:
FXx=∑i{pXxi|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≤FXx≤1 0 FX x 1
  2. FX-∞=0 FX 0 , FX∞=1 FX 1
  3. FXx FX x is non-decreasing as xx increases
  4. Prx1<X≤x2=FXx2−FXx1 x1 X x2 FX x2 FX x1
  5. PrX>x=1−FXx X x 1 FX x
where there is no ambiguity we will often drop the subscript XX and refer to the cdf as Fx F x .

Probability Density Function (pdf)

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

fXx=ddxFXx fX x x FX x (4)
The pdf can also be interpreted in derivative form as δx→0 δ x 0 :
fXxδx=Prx<X≤x+δx=FXx+δx−FXx 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 pXxi pX xi :
fXx=∑i=1MpXxiδx−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, pXxi 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. fXx≥0 fX x 0
  2. ∫-∞∞fXxdx=1 x fX x 1
  3. FXx=∫-∞xfXαdα FX x α x fX α
  4. Prx1<X≤x2=∫x1x2fXαdα x1 X x2 α x1 x2 fX α
As for the cdf, we will often drop the subscript XX and refer simply to fx f x when no confusion can arise.

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