Moment-generating function

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

M_X(t)=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.

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

M_X(\mathbf{t}) = 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 interval around t = 0, the nth moment is given by

E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{\mathrm{d}^n M_X(t)}{\mathrm{d}t^n}\right|_{t=0}.

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!} + \cdots\right) f(x)\,\mathrm{d}x
= 1 + tm_1 + \frac{t^2m_2}{2!} +\cdots,

where mi is the ith moment. MX( − 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).


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

fa:تابع مولد ممان ko:모멘트생성함수 nl:Momentgenererende functiesu:Fungsi nu ngahasilkeun momenhe:פונקציה יוצרת מומנטים


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Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

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