PROBABILITIES Course

Expected value

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The expected value / Espérance noted $\mathbb{E}(X)$, $\mathbb{E}X$, or $\mathbb{E}[X]$ is defined by

\[ \sum_{x_i \in \Omega}^{} x_i \mathbb{P}{(X_w = x_i )} \]

Note: it's the sum of each value by its probability.

Notes

  • $\mathbb{E}(X)$ is also called the (weighted) mean (moyenne (pondérée))
  • $\mathbb{E}(X)$ is also called the first moment (moment d'ordre 1)

Properties

  • Linear: $\mathbb{E}[X, \lambda Y] = \mathbb{E}[Y] + \lambda \mathbb{E}[Y]$
  • Positive: $\mathbb{E}(X) \ge 0$ (if X is a positive v.a.d.)
  • Increasing: $X \ge Y$, $\mathbb{E}(X) \ge \mathbb{E}(Y)$
  • If X,Y independent: $\mathbb{E}[X * Y] = \mathbb{E}[X] * \mathbb{E}[Y]$
  • else: $\mathbb{E}[X * Y] = \sum_{i,j} x_i * y_i * p_{ij}$
  • $\mathbb{E}[c] = c$

Note that $[X,Y]$ means $X$ and $Y$.

If we got an uniform probability, then $\mathbb{E}(X) = \frac{n+1}{2}$.


Chebyshev's inequality

@ \mathbb{P}( |X| \ge a) \le \frac{\mathbb{E}[X^2]}{a^2} @


Jensen

$X$ integrable and $\varphi(x)$ converge

@ \mathbb{E}[\varphi(X)] > \varphi (\mathbb{E}[X]) @

Sometimes, you may see $\mathbb{E}[g(X)] = g(\mathbb{E}[X])$.


Cauchy-Schwarz

@ \mathbb{E}[XY]^2 \le \mathbb{E}[X^2] * \mathbb{E}[Y^2] @


Markov

@ \mathbb{P}( |X| \ge a) \le \frac{\mathbb{E}[ |X| ]}{a} @