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In probability theory and statistics, the moment-generating function of a random variable X is## Calculation

## Significance

## See also

- $M\_X(t)=operatorname\{E\}left(e^\{tX\}right),\; quad\; t\; in\; mathbb\{R\},$

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.

If X has a continuous probability density function f(x) then the moment generating function is given by

- $M\_X(t)\; =\; int\_\{-infty\}^infty\; e^\{tx\}\; f(x),mathrm\{d\}x$

- $=\; int\_\{-infty\}^infty\; left(1+\; tx\; +\; frac\{t^2x^2\}\{2!\}\; +\; cdotsright)\; f(x),mathrm\{d\}x$

- $=\; 1\; +\; tm\_1\; +\; frac\{t^2m\_2\}\{2!\}\; +cdots,$

where $m\_i$ is the ith moment. $M\_X(-t)$ is just the two-sided Laplace transform of f(x).

Regardless of whether the probability distribution is continuous or not, the moment-generating function is given by the Riemann-Stieltjes integral

- $M\_X(t)\; =\; int\_\{-infty\}^infty\; e^\{tx\},dF(x)$

where F is the cumulative distribution function.

If X_{1}, X_{2}, ..., X_{n} is a sequence of independent (and not necessarily identically distributed) random variables, and

- $S\_n\; =\; sum\_\{i=1\}^n\; a\_i\; X\_i,$

where the a_{i} are constants, then the probability density function for S_{n} is the convolution of the probability density functions of each of the X_{i} and the moment-generating function for S_{n} is given by

- $$

For vector-valued random variables X with real components, the moment-generating function is given by

- $M\_X(mathbf\{t\})\; =\; operatorname\{E\}left(e^\{langle\; mathbf\{t\},\; mathbf\{X\}rangle\}right)$

where t is a vector and $langle\; mathbf\{t\}\; ,\; mathbf\{X\}rangle$ is the dot product.

Provided the moment-generating function exists in an open interval around t = 0, the nth moment is given by

- $operatorname\{E\}left(X^nright)=M\_X^\{(n)\}(0)=left.frac\{mathrm\{d\}^n\; M\_X(t)\}\{mathrm\{d\}t^n\}right|\_\{t=0\}.$

If the moment generating function is finite in such an interval, then it uniquely determines a probability distribution.

Related to the moment-generating function are a number of other transforms that are common in probability theory, including the characteristic function and the probability-generating function.

The cumulant-generating function is the logarithm of the moment-generating function.

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Last updated on Tuesday October 07, 2008 at 15:28:04 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday October 07, 2008 at 15:28:04 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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