Expectation

Expectation

The expectation, $\mathbb{E}(X)$, of a discrete random variable, $X$, can be calculated using: $$ \mathbb{E}(X) = \sum_{i=0}^\infty x_i P(X=x_i) $$ The expectation, $\mathbb{E}(Y)$, of a continuous random variable, $Y$, can be calculated using: $$ \mathbb{E}(Y) = \int_{-\infty}^\infty y f_Y(y) \textrm{d}y $$ where $f_Y(y)$ is the probability density function for the random variable. Complex random variables can be constructed by taking functions of simpler random variables. When this is done the expectation, $\mathbb{E}[g(X)]$, of a function, $g(x)$, of a discrete random variable, $X$, can be calculated using: $$ \mathbb{E}[g(X)] = \sum_{i=0}^\infty g(x_i) P(X=x_i) $$ The expectation, $\mathbb{E}[g(Y)]$, of a function, $f(y)$, of a continuous random variable, $Y$, can be calculated using: $$ \mathbb{E}(Y) = \int_{-\infty}^\infty f(y) f_Y(y) \textrm{d}y $$ where $f_Y(y)$ is the probability density function for the random variable. If the function of the random variable is linear and you are being asked to calculate an expection of the form $\mathbb{E}(aX+c)$, where $X$ is a random area and $a$ and $b$ are fixed parameters then the expectation this new random variable can be calulated from the expectation, $\mathbb{E}(X)$ using: \[ \mathbb{E}(aX + c ) = a\mathbb{E}(X) + c \]

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