Expected value is a fundamental concept in the category of probability in mathematics. The expected value of a variable is defined as the average value which is intuitive in long run value by repeatedly performing the experiment which it is representing. Other names for expected value are mathematical expectation, mean, expectation, first moment or EV.

We denote the expected value of a function f (x) by < f (x)> or E [f (x)]

In other words we can say that the expectation of a discrete kind of random variable is simply the weighted average probability of the values that are possible.

The expected value plays an important role in characterization of the probability distribution. It serves as a key aspect since it is one of the location parameter type.

Few properties of expected value are as:

**1)** The expected value of some constant is same as the value of the constant. This implies that E [b] = b, where ‘b is a constant.

E [M + b] = E [M] + b

E [M + N] = E [M] + E [N]

E [b M] = b E [M]

Where ‘b’ is a constant and M and N are random variables.

For a discrete and single variable we define is as follows:

<f (x)> = $\sum_{x}$ f (x) P (x)

Where, P (x) represents the probability density function.

For a continuous and single variable we define it as follows:

<f (x)> = $\int$ f (x) P (x) dx

For discrete and multiple variables we follow

$<f (x_1, x_2,…., x_n)>$ = $\sum_{x_1, x_2,…, x_n} f (x_1, …, x_n) P (x_1, x_2,…, x_n)$

For continuous and multiple variables we follow

$<f (x_1, x_2,…., x_n)>$ = $\int$ $f (x_1, …, x_n) P (x_1, x_2,…, x_n) dx_1 dx_2 …. dx_n$

Below we have discussed three methods to find the expected value:

**The first method follows following steps:**

Any of the above methods can be used to find the expected value of a distribution. Using expected value we can also find the variance and standard deviation of any function.

Let x be a random variable, i.e. up face of a die, so the probability mass function, say g(x) = $\frac{1}{6}$

Now the expected value of x is

E(x) = 1 x $\frac{1}{6}$ + 2 x $\frac{1}{6}$ + 3 x $\frac{1}{6}$ + 4 x $\frac{1}{6}$ + 5 x $\frac{1}{6}$ + 6 x $\frac{1}{6}$

= $\frac{1}{6}$ + $\frac{2}{6}$ + $\frac{3}{6}$ + $\frac{4}{6}$ + $\frac{5}{6}$ + $\frac{6}{6}$

= $\frac{21}{6}$

x |
3 |
4 |
5 |

g(x) | 0.1 | 0.1 | 0.2 |

Find the standard deviation of function.

x |
3 |
4 |
5 |

g(x) | 0.1 | 0.1 | 0.2 |

x-$\mu$ | 1.3 | 2.3 | 3.3 |

(x-$\mu$)$^2$ | 1.69 | 5.29 | 10.89 |

First find expected value from the given data:

E(x) = 3 x 0.1 + 4 x 0.1 + 5 x 0.2 = 1.7

or $\mu$ = 1.7

Now $\sigma^2$ = E(x - $\mu$)$^2$

= 1.69x 0.1 + 5.29 x 0.1 + 10.89 x 0.2

= 0.169 + 0.529 + 2.178

= 2.876

Standard deviation = $\sqrt{ 2.876}$ = 1.70 (approx)