ML Wiki

Expected Value

Expected Value (Математическое ожидание) of a Random Variable X

• is a sum of all possible values from $x_i \in \text{Dom}(X)$ multiplied by their probabilities $p_i$
• denoted $E[X]$ or $M[X]$
• it's often called the center of a Distribution

For discrete random values the formula is

• $E[X] = \sum_{i = 1}^{\infty} x_i p_i$

Mean

Expected Value of $X$ is approximately equal to the mean value of $X$

• $E[X] \approx \bar{X}$

$\bar{X} = x_1 \cfrac{m_1}{n} + x_2 \cfrac{m_2}{n} + ... + x_k \cfrac{m_k}{n}$ where

• $\cfrac{m_i}{n} \approx p_i$ relative frequency of $x_i$

Properties

• $E[C] = C$
• $E[C \cdot X] = C \cdot E[X]$
• $E[X \cdot Y] = E[X] \cdot E[Y]$, if $X$ and $Y$ are independent (proof?)
• and $E[XYZ] = E[X] E[Y] E[Z]$, also if $X$, $Y$ and $Z$ are independent