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What is a probability distribution function?

Module by: Jones Kalunga. E-mail the author

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A mathematical function can be used to model the frequencies and probabilities of occurrences over time. A discrete probability distribution function associates a list of probabilities with each possible value of a discrete random variable. The probability distribution function is thus used to model the probabilities of a discrete random variable and is also known as a probability mass function. The probabilities of a continuous random variable are modelled using continuous distribution functions, also known as probability density functions (pdf's).

The following are particularly important forms of the probability distribution function.

Example 1

This discrete probability density function models experiments that have only two possible outcomes. The probability of success is p and the probability of failure is q=1-p. The pdf models the probability that we will observe r sucesses and n-r failures in a total of n-trials.

Figure 1: Graph of the probability distribution function and the cumulative probability distribution function (redrawn from http://www.engr.udayton.edu/faculty/mdaniels/htm315/Functions.htm using matlab)
Figure 1 (pdf1.jpg)

Exercise 1

From the example above, what is the probability that in 20-trials there are exactly six successes?

Solution

The probability that there are exactly six successes is 0.04

References:

  1. Random Variables and their Probability Density and Distribution Functions, http://www.engr.udayton.edu/faculty/mdaniels/htm315/Functions.htm (last accessed February 2006)
  2. NCAR Advanced Study Program http://www.asp.ucar.edu (last accessed February 2006)

Co-Author: Mookho Tsilo

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