class04-2-handouts

class04-2-handouts - Test topics Distributions of functions...

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Unformatted text preview: Test topics Distributions of functions of random variables Method of distribution functions Alternative method of distribution functions Method of transformations Method of moment generating functions Important sampling distributions Sums and averages of normals Sums of squares of standard normals Ratios of standard normals to square roots of 2 s divided by their d.o.f. Ratios of 2 s divided by their d.o.f Central limit theorem Asymptotic distributions of sums and averages of random variables Continuity correction Estimators Method of moments Maximum likelihood Properties of estimators Bias Mean square error New distributions Beta Geometric Jarad Niemi (UCSB) Midterm Review I 21 April 2010 1 / 16 Distributions of functions of random variables Method of distribution functions Definition Let U = h ( Y 1 , Y 2 ,..., Y n ). The method of distribution functions is 1 Find the region U = u in the ( y 1 , y 2 ,..., y n ) space. 2 Find the region U u . 3 Find F u ( u ) = P ( U u ) by integrating f Y ( y 1 , y 2 ,..., y n ) over the region U u . 4 Find the density function f U ( u ) by differentiating F U ( u ). Thus, f U ( u ) = dF U ( u ) / du . Jarad Niemi (UCSB) Midterm Review I 21 April 2010 2 / 16 Alternative Distribution Function Method Alternative distribution function method Definition Let Y have cumulative distribution function (cdf) F Y ( y ), let U = h ( Y ) and let Y and U be the support of Y and U , respectively. The alternative method of distribution functions is 1 If h ( ) is an increasing function on Y , F U ( u ) = F Y ( h- 1 ( u )) for u U ....
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class04-2-handouts - Test topics Distributions of functions...

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