For continuous random variables, double summation replaced by a double integral in the definition.
Definition
If \(X\) and \(Y\) are continuous random variables with joint probability density function \(f(x,y)\), then the expected value of a function of the variables, \(g(X,Y)\), is defined to be
\[ E[g(X,Y)] \;=\; \int_{-\infty}^{\infty}{\int_{-\infty}^{\infty} {g(x,y) \times f(x,y)} \;dx} \;dy \]Each possible value of \(g(x,y)\) is "weighted" by its probability density — the most likely values of \((x,y)\) contribute most to the expected value.
Example
A point is randomly selected within a unit square. What is the expected area of the rectangle with it and the origin as corners? What is its variance? |
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(Solved in full version)