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Probability: Difference between revisions
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The CDF is the prefix sum of the PMF or the integral of the PDF. Likewise, the PDF is the derivative of the CDF. | The CDF is the prefix sum of the PMF or the integral of the PDF. Likewise, the PDF is the derivative of the CDF. | ||
==Expectation | ==Expectation and Variance== | ||
Some definitions and properties. | Some definitions and properties. | ||
===Definitions=== | ===Definitions=== | ||
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* <math>(n-1)S^2 / \sigma^2 \sim \chi^2(n-1)</math> | * <math>(n-1)S^2 / \sigma^2 \sim \chi^2(n-1)</math> | ||
=== | ==Moments and Moment Generating Functions== | ||
===Definitions=== | |||
{{main | Wikipedia: Moment (mathematics) | Wikipedia: Central moment | Wikipedia: Moment-generating function}} | {{main | Wikipedia: Moment (mathematics) | Wikipedia: Central moment | Wikipedia: Moment-generating function}} | ||
* <math>E(X^n)</math> the n'th moment | * <math>E(X^n)</math> the n'th moment | ||
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Additionally, ''skew'' is the third standardized moment and ''kurtosis'' is the fourth standardized moment. | Additionally, ''skew'' is the third standardized moment and ''kurtosis'' is the fourth standardized moment. | ||
===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>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 | * The MGF, if it exists, uniquely defines the distribution. | ||
* The | * 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=== | ||