Lets first define a few terms.

**Discrete Random Variable** - This is a variable that can only take on particular values. For example heads or tails of a coin, it makes no sense to talk about 3/4's heads or 1/8 tails. Another example, the integers 1 through 6 are possible values when rolling a die. There is no way to get 4.3 on a standard die.

**Continuous Random Variable** - This is in contrast to the definition above only in that the variable can take on any value in a range however, the range is generally limited by the context of the problem. For example, the height of human can take on virtually any value within a range (say 1 meter to 2.5 meters). It would be very odd to find that every person's height was an integer value of centimeters!

#### Discrete Probability Distributions

The probability of a outcome of an event occurring is between 0 and 1. If an event has *n* possible outcomes then the probability of each outcome can be denoted as $p_1,p_2,...,p_n$. Then the sum of the probabilities must equal 1. Or in equation form:

If the sum of the probabilities does not equal 1, this implies that either a mistake has been made or not all possible outcomes have been included. All these individual values of probabilities make up a "probability distribution." Probability distributions can be represented three separate ways:

##### 1 Table Form

The table below shows the probability *P(X)* of seeing *x* number of heads after flipping a coin 4 times.

x |
0 | 1 | 2 | 3 |

P(x) |
0.125 | 0.375 | 0.375 | 0.125 |

##### 2 Graphically

One way to visualize a random variable is to create a graph of all the possible values that the variable can take on. For a discrete variable this could look like a bar graph, with the height of the bar indicating the probability of that value. The bar graph below shows the probability of rolling each number on standard six-sided die.

##### 3 Functional Form

The probability that a variable takes on a particular value is denoted by:

(2)Where *X* is the variable and *a* is the value in question. For example, for a standard six-sided die the probability that a 3 is rolled is $P(X=3)=\frac{1}{6}$

#### Expectation Value

Mathematically the expectation value can be written as:

(3)Where *n* is the number of possible outcomes, $x_i$ is the value of an outcome and $p_i$ is the probability that $x_i$ occurs.

##### Side Note

Dan Meyer has a nice post about expectation value. If you can follow what he's getting at your probably doing well.

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