## Expectation-probability formulas

### 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

### 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:

$$E(X)= \int\limits_0^2 xf(x)dx$$
$$E(X)= \int\limits_0^2 \frac 38 x^3 dx$$
$$E(X)= \frac {3x^4}{32} |_0^2$$
$$E(X)= \frac 32$$