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:

(1)

\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.

+ Another Example

Another example would be the probability distribution of flipping a coin twice. We can express the probability in terms of the number of heads that would be seen. Since there are four possible outcomes (HH, HT, TH and TT) each with equal probability (I'll let you work that out). The probability distribution look like the one below.

3 Functional Form

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

(2)

\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:

(3)

\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.

+ Click for more of an explanation

This is a tough concept to explain (I didn't do a brilliant job) and maybe harder to understand. The expectation value is a way of giving a "weight" to each possible outcome. Lets use the example of rolling a six-sided die. Each outcome has an equal probability, but if I asked you to place a bet or make a guess about the most likely value that would occur what would you say…?

Another way of looking at this, is to ask what would the average value of the outcome be if the die was rolled several times? Since all of the outcomes have equal probability, the average value would come out to be:

(4)

\begin{align} \frac{1+2+3+4+5+6}{6}=3.5 \end{align}

Which is just the average of all the values on the faces of the die. This can also be rewritten as

(5)

\begin{align} 1 \left( \frac{1}{6} \right ) + 2 \left( \frac{1}{6} \right ) + 3 \left( \frac{1}{6} \right ) + 4 \left( \frac{1}{6} \right ) + 5 \left( \frac{1}{6} \right ) + 6 \left( \frac{1}{6} \right ) = 3.5 \end{align}

This can be interpreted as the value of each outcome ($x_i)$ multiplied by the probability ($p_i$) of that outcome. Each term in the summation is $x_i p_i$.

In general the expectation value is the sum of the product of the value of an outcome and the probability. Or

(6)

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

The IB likes the notation of $E(X)$ for the expectation value.

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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