Define mathematical Expectation of a discrete random variable.

1. Random variable:

Let $S$ be the sample space associated with a given random experiment. Then, a real valued function $X$ which assigns to each event $w\in S$ to a unique real number $X\left(w\right)$ is called a random variable.

2. Probability distribution of a random variable:

If a random variable $X$ takes values $x_1, x_2, ...., x_n$ with respective probabilities $p_1, p_2,.... , p_n$ then

$p_i\ge 0$ and $\sum _{}{p}_i=1$ is known as the probability distribution of $X$.

3. Mathematical expectation:

(i) Expectation of discrete random variable:

$E\left(X\right)=\sum _{i=1}^n{x}_i{p}_i$

(ii) Variance of discrete random variable:

$\mathrm{Var}\left(X\right)=E\left(X^2\right)-(E\left(X\right){)}^2$

4. Addition and multiplication theorems on expectations:

(i) Addition theorem:

$E\left(X+Y\right)=E\left(X\right)+E\left(Y\right)$

(ii) Multiplication theorem:

$E\left(XY\right)=E\left(X\right)E\left(Y\right)$