For a continuous random variable X with density f, the mean is the probability-weighted average of its possible values. mu=E(X)= _all xxf(x),dx, provided the expectation is finite. For a function g, apply the function to each value before weighting by the density: E[g(X)]= _all xg(x)f(x),dx, provided this expectation is finite. In particular, E(X2)= _all xx2f(x),dx. This is generally different from [E(X)]2. Integrate over the support of the density. For a piecewise density, add the integrals over its separate intervals.

For a continuous random variable \(X\) with density \(\mathrm{f}\), the mean is the probability-weighted average of its possible values.

\(\mu=\mathrm{E}(X)=\int_{\text{all }x}x\mathrm{f}(x)\,\mathrm{d}x\), provided the expectation is finite.

For a function \(\mathrm{g}\), apply the function to each value before weighting by the density:

\(\mathrm{E}[\mathrm{g}(X)]=\int_{\text{all }x}\mathrm{g}(x)\mathrm{f}(x)\,\mathrm{d}x\), provided this expectation is finite.

In particular, \(\mathrm{E}(X^2)=\int_{\text{all }x}x^2\mathrm{f}(x)\,\mathrm{d}x\). This is generally different from \([\mathrm{E}(X)]^2\).

Integrate over the support of the density. For a piecewise density, add the integrals over its separate intervals.

Continuous Random Variables

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Continuous Expectation

Continuous Expectation