### Math 225 Course Notes

### Section 4.2:
Probability Distributions of Discrete Random Variables

The distribution of
discrete random variables can be displayed
in a table
or with a formula.
The sum of the probabilities of the possible values is one,
and all probabilities are between 0 and 1.
Cumulative probability distributions
are an alternative way to display the distributions of random variables.

**Example:**
x | 3 | 4 | 5 | 6
-------------------------------
P(X=x) | .2 | .3 | .4 | .1

Capital letters represent random variables.
Small letters represent the possible values of random variables.
In this example, the random variable X has four possible values;
3, 4, 5, and 6.

The probability that the X is 4 equals .3,
or P(X = 4) = 0.3.

The sum of all the probabilities of all the possible values is one,
and all probabilities are between zero and one.

**Example:**
Sometimes, instead of writing out the whole table,
the distribution is specified by a formula.

P( X = x ) = x/10 for x = 1,2,3, or 4

In a table, this would be
x | 1 | 2 | 3 | 4
--------------------------------------
P(X=x) | 1/10 | 2/10 | 3/10 | 4/10

The cumulative probability distribution function tells P(X <= x).
In the two previous examples,
here are the cumulative probability distributions.

**Example 1:**

x | P(X <= x)
----------------
3 | .2
4 | .5
5 | .9
6 | 1.0

**Example 2:**
x | P(X <= x)
----------------
1 | .1
2 | .3
3 | .6
4 | 1.0

Last modified: Feb 5, 1996

Bret Larget,
larget@mathcs.duq.edu