The expected value of a [[Random Variable]] can be interpreted as the average outcome over a large number of [[Independence of Random Variables|independent]] draws from that random variable. For the calculation we iterate over all possible values $x$, and weight the respective $x$ with its probability of occurrence $p_X(x)$. ^52e656 $ \begin{align} \mathbb E[X]&= \sum_x x*p_X(x) \quad &&\text{(discrete)} \\[4pt] \mathbb E[X]&= \int_{-\infty}^\infty x*f_X(x)\, dx \quad &&\text{(continuous)} \end{align} $ ^32925b ## Properties of Expectation **Property 1:** If $X \ge 0$ then $\mathbb E[X] \ge 0$ - *Description:* If all $x_i$ are $\ge0$, then the expectation has to be $\ge 0$ as well. - *Proof:* Since probabilities are always $\ge 0$ by [[Probability Axioms#Axioms|axiom]], and in this case $x_i \ge0$, we are only multiplying and summing non-negative terms. $ \mathbb E[X] = \sum_x x* \underbrace{p_X(x)}_{\ge0} $ **Property 2:** If $a \le X \le b$ then $a \le \mathbb E[X] \le b$ - *Description:* If all $x_i$ are between $[a,b]$, then the expectation is within these bounds as well. - *Proof:* Assume the edge case, where all $x_i=a$, where $a$ is the lower bound of the range. $ \begin{align} \mathbb E[X] &= \sum_x x*p_X(x) \\ &=\sum_x a*p_X(x) \\ &=a*\underbrace{\sum_x p_X(x)}_{=1}=a \end{align} $ **Property 3:** If $X$ is a constant then $\mathbb E[X] = x$ - If there is only of possible outcome $x$ from the r.v. $X$, then the expectation is $x$ as well. **Property 4:** If $Y=g(X)$ then the expectation of $Y$can be evaluated in 2 ways. - *Directly:* Iterating over the possible values of $Y$. $ \mathbb E[Y] = \sum_y y*p_Y(y) $ - *Indirectly:* Iterating over the possible values of $X$ transformed by $g(X)$. $ \mathbb E[Y] = \mathbb E[g(X)] = \sum_x g(X)*p_X(x) $