Expectation value and Variance

Defintion of Expectation

Expected Value( $E (X)$ ) to denote that this is an average(or mean) for a random variable$$E(X)=x_{i} p_{i}$$
Whera $x_{i}$ is value of outcome $i$ and $p_{i}$ is the probability of outcome $i$

Calculation rules

$E (X + Y) = E (X) + E (Y)$

Definition of Variance

Like what it called, variance( $V a r (X)$ ) measures the variability during the process and in probability with formula$$Var(X)=E[(X-\mu)^{2}]= E(X^{2})-\mu^{2}=\sum (x_i)^2p_{i}-\mu^2$$

Calculation rules

We need to assume X and Y are independent of each other

Addition : $V a r (X + Y) = V a r (X) + V a r (Y)$
Substract: $V a r (X - Y) = V a r (X) + V a r (Y)$
scalar multiplcation $V a r (c X) = c^{2} V a r (X)$