# Expected Value

In a
probability distribution
, the
**
weighted average
**
of possible values of a random variable, with weights given by their respective theoretical probabilities, is known as the
**
expected value
**
, usually represented by
$E\left(x\right)$
.

The expected value informs about what to expect in an experiment "in the long run", after many trials. In most of the cases, there could be no such value in the sample space.

The weighted average formula for expected value is given by multiplying each possible value for the random variable by the probability that the random variable takes that value, and summing all these products. It can be written as

$E\left(x\right)={\displaystyle \sum {x}_{i}P\left({x}_{i}\right)}$ ,

where ${x}_{i}$ covers all possible values for the random variable, and $P\left({x}_{i}\right)$ is the respective theoretical probability.

$E\left(x\right)$
is also called as mean of the probability distribution because it tells what to expect in the "
*
long run
*
"- that is, after many trials.

**
Example:
**

When you roll a die, you will be paid $\$1$ for odd number and $\$2$ for even number. Find the expected value of money you get for one roll of the die.

The sample space of the experiment is $\left\{1,2,3,4,5,6\right\}$ .

The table illustrates the probability distribution for a single roll of a die and the amount that will be paid for each outcome.

Roll ( $x$ ) | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |

Probability | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ |

Amount ($) | $1$ | $2$ | $1$ | $2$ | $1$ | $2$ |

Use the weighted average formula.

$\begin{array}{l}E\left(x\right)=1\left(\frac{1}{6}\right)+2\left(\frac{1}{6}\right)+1\left(\frac{1}{6}\right)+2\left(\frac{1}{6}\right)+1\left(\frac{1}{6}\right)+2\left(\frac{1}{6}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{6}+\frac{2}{6}+\frac{1}{6}+\frac{2}{6}+\frac{1}{6}+\frac{2}{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{9}{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1.5\end{array}$

So, the expected value is $\$1.50$ . In other words, on average, you get $\$1.50$ per roll.