Difference between revisions of "Expectation (math)"
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| − | The '''expectation''' of continuously distributed variable <math>X</math> with [[probability density function]] | + | The mathematical '''expectation''' of a continuously distributed random variable <math>X</math> with [[probability density function]] |
<math>f(x)</math> is | <math>f(x)</math> is | ||
:<math> | :<math> | ||
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f(x)dx. | f(x)dx. | ||
</math> | </math> | ||
| − | + | The expectation is also known as the [[mean]] of <math>X</math>. | |
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| − | The expectation is also the [[mean]] of | ||
The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is | The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is | ||
| Line 25: | Line 17: | ||
</math> | </math> | ||
| − | [[ | + | For a discretely distributed random variable <math>X</math> with [[probability mass function]] |
| + | <math>p_{k}</math> it is | ||
| + | :<math> | ||
| + | \mbox{E}[X]=\sum_{k} p_{k}x_{k}. | ||
| + | </math> | ||
| + | |||
| + | [[Category:Probability and Statistics]] | ||
Latest revision as of 12:16, July 13, 2016
The mathematical expectation of a continuously distributed random variable <math>X</math> with probability density function <math>f(x)</math> is
- <math>
\mbox{E}[X] =\int\limits_{-\infty}^\infty x f(x)dx. </math> The expectation is also known as the mean of <math>X</math>.
The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is
- <math>
\mbox{E}[g(X)] =\int\limits_{-\infty}^\infty g(x) f(x)dx. </math>
For a discretely distributed random variable <math>X</math> with probability mass function <math>p_{k}</math> it is
- <math>
\mbox{E}[X]=\sum_{k} p_{k}x_{k}. </math>