expected value

Topic

• probability
• descriptive statistics

The expected value represents the long-run average of a random variable’s possible outcomes. It’s calculated as the sum of all possible values weighted by their probabilities.

For discrete random variables:

$$E[X]=\sum x_i\cdot P(x_i)$$

For continuous variables:

$$E[X]={\int }_{-\infty }^{\infty }xf(x)dx$$

Key properties:

• Linearity: $E[aX+b]=aE[X]+b$
• Additivity: $E[X+Y]=E[X]+E[Y]$

The sample mean

$$\bar{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$$

estimates the population expected value $\mu$ from observed data

from numpy import array, mean
 
M = array([
[1,2,3,4,5,6],
[1,2,3,4,5,6]])
 
col_mean = mean(M, axis=0)
row_mean = mean(M, axis=1)

Related

• variance