By Olive D.J.

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**Sample text**

Yn) ∂y1 · · · ∂yn wherever the partial derivative exists. 5. , yik are held ﬁxed. , yk are held ﬁxed. 2) y2 where y1 is held ﬁxed. 3) y1 where y2 is held ﬁxed. 1. For n = 2, double integrals are used to ﬁnd marginal pdfs (deﬁned below) and to show that the joint pdf integrates to 1. If the CHAPTER 2. MULTIVARIATE DISTRIBUTIONS 29 region of integration Ω is bounded on top by the function y2 = φT (y1), on the bottom by the function y2 = φB (y1) and to the left and right by the lines f(y1 , y2)dy1 dy2 = f(y1 , y2)dy2 dy2 = y1 = a and y2 = b then Ω Ω b φT (y1 ) φB (y1 ) a f(y1 , y2)dy2 dy1 .

Yn are independent Cauchy C(µi , σi ), then n n (ai + bi Yi ) ∼ C( i=1 n |bi |σi ). (ai + bi µi ), i=1 n i=1 i=1 Yi ∼ BIN(nk, ρ). CHAPTER 2. , Yn are iid C(µ, σ), then Y ∼ C(µ, σ). c) Denote a chi–square χ2p random variable by χ2(p). , Yn are independent chi–square χ2pi , then n n 2 Yi ∼ χ ( i=1 pi ). , Yn are iid χ2p , then n Yi ∼ χ2np . , Yn are iid exponential EXP(λ), then n Yi ∼ G(n, λ). , Yn are independent Gamma G(νi , λ) then n n Yi ∼ G( i=1 νi , λ). , Yn are iid G(ν, λ), then n Yi ∼ G(nν, λ).

An exception is the multivariate normal distribution and the elliptically contoured distributions. ) For independent random variables, the joint cdf is the product of the marginal cdfs, the joint pmf is the product of the marginal pmfs, and joint pdf is the product of the marginal pdfs. 9. , yn. , yn. If the random variables are not independent, then they are dependent. In particular random variables Y1 and Y2 are independent, written Y1 Y2 , if either of the following conditions holds. i) F (y1, y2 ) = FY1 (y1)FY2 (y2) ∀y1, y2.

### A Course in Statistical Theory by Olive D.J.

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