Discrete Variables

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:

\begin{align} p_1+p_2+...+p_n = \sum_{i=1}^n p_i = 1 \end{align}

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:

\begin{equation} P(X=a) \end{equation}

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:

\begin{align} E(X)= \sum_{i=1}^{n}x_i p_i \end{align}

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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