capsizing moment

Moment-generating function

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

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 X1, X2, ..., Xn 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 ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi and the moment-generating function for Sn is given by

M_{S_n}(t)=M_{X_1}(a_1t)M_{X_2}(a_2t)cdots M_{X_n}(a_nt).

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.

See also

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