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Probability: Difference between revisions
→Moment Generating Functions
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===Moment Generating Functions=== | ===Moment Generating Functions=== | ||
To compute moments, we can use a moment generating function (MGF): | To compute moments, we can use a moment generating function (MGF): | ||
<math>M_X(t) = E(e^{tX})</math> | <math display="block">M_X(t) = E(e^{tX})</math> | ||
With the MGF, we can get any order moments by taking n derivatives and setting <math display="inline">t=0</math>. | With the MGF, we can get any order moments by taking n derivatives and setting <math display="inline">t=0</math>. | ||
; Notes | ; Notes | ||
* The MGF, if it exists, uniquely defines the distribution. | * The MGF, if it exists, uniquely defines the distribution. | ||
* The MGF of <math>X+Y</math> is <math>MGF_{X+Y}(t) = E(e^{t(X+Y)})=E(e^{tX})E(e^{tY}) = MGF_X(t) * MGF_Y(t)</math> | * The MGF of <math>X+Y</math> is <math>MGF_{X+Y}(t) = E(e^{t(X+Y)})=E(e^{tX})E(e^{tY}) = MGF_X(t) * MGF_Y(t)</math> | ||
===Characteristic function=== | ===Characteristic function=== | ||