Math Formulas || Probability || Expectation

Expectation of Continuous Random Variable :

$E(X)~=\mu~=\large\int^\infty_{-\infty} ~xP(x)~dx \\$

where,

$ \mu~=$E(X) is the expectation value of the continuous random variable X

$x$ is the value of the continuous random variable X

P(x) is the probability density function

Expectation of Discrete Random Variable :

$E(X)~=~\sum\limits_{i=1}^{n}~x_i p_i $

E(X) is the expectation value of the discrete random variable X

$x_i$ =Outcomes

$p_i$ = Probabilities

Properties of Expectation:

1. Linearity

When a is constant and X,Y are random variables:

E(aX) = aE(X)

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

2. Constant

When c is constant:

E(c) = c

3. Product

When X and Y are independent random variables:

E(X ·Y) = E(X) · E(Y)