Random Variables

Expected Value

Expected value is the long-run average of a random variable:

E(X) = Σ x·P(x)

Multiply each value by its probability and add the terms.

Example

Multiply each value by its probability, then add.

highlighted = computed this step

Step 1 — Distribution

List the distribution and leave products blank.

xPterm116□216□316□416□516□616□\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \square \\ 2 & \frac{1}{6} & \square \\ 3 & \frac{1}{6} & \square \\ 4 & \frac{1}{6} & \square \\ 5 & \frac{1}{6} & \square \\ 6 & \frac{1}{6} & \square\end{array}

Step 2 — Term for x=1

Multiply x times probability for this row.

xPterm11616216□316□416□516□616□\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \hlmath{\frac{1}{6}} \\ 2 & \frac{1}{6} & \square \\ 3 & \frac{1}{6} & \square \\ 4 & \frac{1}{6} & \square \\ 5 & \frac{1}{6} & \square \\ 6 & \frac{1}{6} & \square\end{array}

Step 3 — Term for x=2

Multiply x times probability for this row.

xPterm1161621613316□416□516□616□\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \frac{1}{6} \\ 2 & \frac{1}{6} & \hlmath{\frac{1}{3}} \\ 3 & \frac{1}{6} & \square \\ 4 & \frac{1}{6} & \square \\ 5 & \frac{1}{6} & \square \\ 6 & \frac{1}{6} & \square\end{array}

Step 4 — Term for x=3

Multiply x times probability for this row.

xPterm116162161331612416□516□616□\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \frac{1}{6} \\ 2 & \frac{1}{6} & \frac{1}{3} \\ 3 & \frac{1}{6} & \hlmath{\frac{1}{2}} \\ 4 & \frac{1}{6} & \square \\ 5 & \frac{1}{6} & \square \\ 6 & \frac{1}{6} & \square\end{array}

Step 5 — Term for x=4

Multiply x times probability for this row.

xPterm11616216133161241623516□616□\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \frac{1}{6} \\ 2 & \frac{1}{6} & \frac{1}{3} \\ 3 & \frac{1}{6} & \frac{1}{2} \\ 4 & \frac{1}{6} & \hlmath{\frac{2}{3}} \\ 5 & \frac{1}{6} & \square \\ 6 & \frac{1}{6} & \square\end{array}

Step 6 — Term for x=5

Multiply x times probability for this row.

xPterm1161621613316124162351656616□\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \frac{1}{6} \\ 2 & \frac{1}{6} & \frac{1}{3} \\ 3 & \frac{1}{6} & \frac{1}{2} \\ 4 & \frac{1}{6} & \frac{2}{3} \\ 5 & \frac{1}{6} & \hlmath{\frac{5}{6}} \\ 6 & \frac{1}{6} & \square\end{array}

Step 7 — Term for x=6

Multiply x times probability for this row.

xPterm11616216133161241623516566161\begin{array}{ccc}\text{x} & \text{P} & \text{term} \\ 1 & \frac{1}{6} & \frac{1}{6} \\ 2 & \frac{1}{6} & \frac{1}{3} \\ 3 & \frac{1}{6} & \frac{1}{2} \\ 4 & \frac{1}{6} & \frac{2}{3} \\ 5 & \frac{1}{6} & \frac{5}{6} \\ 6 & \frac{1}{6} & \hlmath{1}\end{array}

Step 8 — Expected value

Add the weighted products for expected value.

E[X]=16+13+12+23+56+1=72E[X]= \frac{1}{6} + \frac{1}{3} + \frac{1}{2} + \frac{2}{3} + \frac{5}{6} + 1 = \hlmath{\frac{7}{2}}
expected-value E(X) = Σ x·P(x)