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Probability: Difference between revisions
→Expectation and Variance
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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 and | ==Expectation, Variance, and Moments== | ||
Some definitions and properties. | Some definitions and properties. | ||
===Definitions=== | ===Definitions=== | ||
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* <math>E(X) = \sum_S xp(x)</math> or <math>\int_S xp(x)dx</math> | * <math>E(X) = \sum_S xp(x)</math> or <math>\int_S xp(x)dx</math> | ||
* <math>Var(X) = E[(X-E(X))^2] = E(X^2) - (E(X))^2</math> | * <math>Var(X) = E[(X-E(X))^2] = E(X^2) - (E(X))^2</math> | ||
===Total Expection=== | ===Total Expection=== | ||
<math>E_{X}(X) = E_{Y}(E_{X|Y}(X|Y))</math><br> | <math>E_{X}(X) = E_{Y}(E_{X|Y}(X|Y))</math><br> | ||
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{{hidden | Proof | | {{hidden | Proof | | ||
<math> | <math> | ||
E(X) = \int_S | E(X) = \int_S x p(x)dx | ||
= \int_x x \int_y p(x,y)dy dx | = \int_x x \int_y p(x,y)dy dx | ||
= \int_x x \int_y p(x|y)p(y)dy dx | = \int_x x \int_y p(x|y)p(y)dy dx | ||
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This one is not used as often on tests as total expectation | This one is not used as often on tests as total expectation | ||
{{hidden | Proof | | {{hidden | Proof | | ||
}} | }} | ||
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* <math>\bar{X} \sim N(\mu, \sigma^2 / n)</math> | * <math>\bar{X} \sim N(\mu, \sigma^2 / n)</math> | ||
* <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=== | |||
{{main | Wikipedia: Moment (mathematics) | Wikipedia: Central moment | Wikipedia: Moment-generating function}} | |||
* <math>E(X^n)</math> the n'th moment | |||
* <math>E((X-\mu)^n)</math> the n'th central moment | |||
* <math>E(((X-\mu) / \sigma)^n)</math> the n'th standardized moment | |||
Expectation is the first moment and variance is the second central moment.<br> | |||
Additionally, ''skew'' is the third standardized moment and ''kurtosis'' is the fourth standardized moment. | |||
To compute moments, we can use a moment generating function (MGF): | |||
<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>. | |||
==Moments and Moment Generating Functions== | ==Moments and Moment Generating Functions== | ||