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

#### Key Concepts

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.

#### Discrete Distributions

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
```

#### Cumulative Distributions

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
```