Difference between revisions of "Expectation (math)"
m (Expectation moved to Expectation (math): esoteric) |
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| − | The '''expectation''' of | + | The '''expectation''' of continuously distributed 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> | ||
| − | + | ||
| + | For a discretely distributed 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> | ||
| + | |||
| + | |||
| + | The expectation is also the [[mean]] of random variable <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> is | ||
Revision as of 23:53, January 18, 2009
The expectation of continuously distributed 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)
For a discretely distributed variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} with probability mass function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{k}} it is
![{\displaystyle {\mbox{E}}[X]=\sum _{k}p_{k}x_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce0ff23320ca127f3d35ecea767fe6d6e5fce44)
The expectation is also the mean of random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X}
.
The expectation with respect to some function 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)