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By Olive D.J.

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Yn) > 0}. 1. , yn). , 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 .

For example, let n = 3 and ai = i for i = 1, 2, 3. Then ni=1 ai = a1a2 a3 = (1)(2)(3) = 6. 2. , Yn with joint pdf or pmf f are dependent if their support Y is not a cross product. In particular, Y1 and Y2 are dependent if Y does not have the form Y = Y1 × Y2 . , yn) = h1 (y1)h2 (y2) · · · hn (yn ) for all y ∈ Y where hi is a positive function of yi alone. In particular, if Y = Y1 × Y2 , then Y1 Y2 iff f(y1 , y2) = h1(y1 )h2 (y2) for all (y1, y2 ) ∈ Y where hi (yi ) > 0 for yi ∈ Yi and i = 1, 2. c) If discrete Y1 and Y2 have rectangular support given by a table, find the row and column sums.

23. , yn ). Let fY i (y i ) be the marginal pdf or pmf of Y i . , yn ) = fY 1 (y 1 ) · · · fY n (y n ) = fY i (y i ). 7. 18. , n. , n. Then the random vectors X i = hi (Y i ) are independent. There are three important special cases. i) If pji = 1 so that each hi is a real valued function, then the random variables Xi = hi (Y i ) are independent. , Xn are independent. X. , Xm ) and assume that Y h(Y ) is a vector valued function of Y alone and if g(X) is a vector valued function of X alone, then h(Y ) and g(X) are independent random vectors.

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A Course in Statistical Theory by Olive D.J.


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