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==Moments and Moment Generating Functions== | |||
===Definitions=== | |||
We call <math>E(X^i)</math> the i'th moment of <math>X</math>.<br> | |||
We call <math>E(|X - E(X)|^i)</math> the i'th central moment of <math>X</math>.<br> | |||
Therefore the mean is the first moment and the variance is the second central moment. | |||
===Moment Generating Functions=== | |||
<math>E(e^{tX})</math><br> | |||
We call this the moment generating function (mgf).<br> | |||
We can differentiate it with respect to <math>t</math> and set <math>t=0</math> to get the higher moments. | |||
; Notes | |||
* The mgf, if it exists, uniquely defines the distribution. | |||
* The mgf of <math>X+Y</math> is <math>E(e^{t(X+Y)})=E(e^{t(X)})E(e^{t(Y)})</math> | |||
===Characteristic function=== | |||
==Convergence== | ==Convergence== | ||