Functions of Normal Random Variables

Wakefield, Jon

doi:10.1007/978-1-4419-0925-1_17

Jon Wakefield²

Part of the book series: Springer Series in Statistics ((SSS))

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Abstract

If Y _j∼N(0,1), j=1,…,J, with Y ₁,…,Y _J independent, then

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Keywords

Normal Random Variables

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

If Y _j ∼ N(0, 1), j = 1, …, J, with Y ₁, …, Y _J independent, then

$$Z ={ \sum \nolimits }_{j=1}^{J}{Y }_{ j}^{2} \sim {\chi }_{ J}^{2}, $$

(E.1)

achi-squared distribution with J degrees of freedom and $\mbox{ E}[Z] = J,\mbox{ var}(Z) = 2J$.

If X ∼ N(0, 1), Y ∼ χ_d ², with X and Y independent, then

$$\frac{X} {{(Y/d)}^{1/2}} \sim \mbox{ T}(0,1,d), $$

(E.2)

aStudent’s t distribution with d degrees of freedom.

If U ∼ χ_J ² and V ∼ χ_K ², with U and V independent, then

$$\frac{U/J} {V/K} \sim \mbox{ F}(J,K), $$

(E.3)

theF distribution with J, K degrees of freedom.

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Departments of Statistics and Biostatistics, University of Washington, Seattle, Washington, USA
Jon Wakefield

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Wakefield, J. (2012). Functions of Normal Random Variables. In: Bayesian and Frequentist Regression Methods. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0925-1_17

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DOI: https://doi.org/10.1007/978-1-4419-0925-1_17
Published: 04 November 2012
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