Expectation
In probability and statistics, the expectation or expected value, is the weighted average value of a random variable.
Expectation of continuous random variable:
$$ E(X)= \int\limits_{-\infty}^{\infty} x P(x) dx $$
Where,
- 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:
$$ E(X)=\sum_i x_i P(x) $$
Where,
- 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 mass function of X
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:
$$ E(X \cdot Y) = E(X) \cdot E(Y) $$
Example:
Let X have range [0,2] and density (3x2)/8. Find E(X)
Solution:
$$ E(X)= \int\limits_0^2 xf(x)dx $$
$$ E(X)= \int\limits_0^2 \frac 38 x^3 dx $$
$$ E(X)= \frac {3x^4}{32} |_0^2 $$
$$ E(X)= \frac 32 $$