Difference between revisions of "Expectation (math)"
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| − | The '''expectation''' of random 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>. | |
| − | The expectation with respect to some function <math>g(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 |
:<math> | :<math> | ||
\mbox{E}[g(X)] | \mbox{E}[g(X)] | ||
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</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 with probability density function is
![{\displaystyle {\mbox{E}}[X]=\int \limits _{-\infty }^{\infty }xf(x)dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc4a1723078d7a453b6541d844edf798c377f05)
The expectation is also known as the mean of .
The expectation with respect to some function where is distributed according to is
![{\displaystyle {\mbox{E}}[g(X)]=\int \limits _{-\infty }^{\infty }g(x)f(x)dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46ed321df16947ba8913e344b3bedd17e451005f)
For a discretely distributed random variable with probability mass function it is
![{\displaystyle {\mbox{E}}[X]=\sum _{k}p_{k}x_{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3680c282419a8c960e43e5541dd016022f625617)