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


See Also

Sources

  • Гмурман В.Е., Теория вероятностей и математическая статистика -- 9-е издание. М.: Высш. шк., 2003.

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