By Olive D.J.

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Let Y1 = X1 and Y2 = X1 + X2 . The important step is ﬁnding the support Y of (Y1 , Y2 ) from the support of (X1 , X2 ) = X = {(x1, x2)|0 < x1 < x2 < ∞}. −1 Now x1 = y1 = t−1 1 (y1 , y2 ) and x2 = y2 − y1 = t2 (y1 , y2 ). Hence x1 < x2 implies y1 < y2 − y1 or 2y1 < y2, and Y = {(y1 , y2)|0 < 2y1 < y2 }. Now ∂t−1 ∂t−1 1 1 = 1, = 0, ∂y1 ∂y2 ∂t−1 ∂t−1 2 2 = −1, = 1, ∂y1 ∂y2 and the Jacobian J= 1 0 −1 1 = 1. CHAPTER 2. MULTIVARIATE DISTRIBUTIONS 46 Hence |J | = 1. Using indicators, gX1 ,X2 (x1, x2 ) = 2e−(x1 +x2 ) I(0 < x1 < x2 < ∞), and fY1 ,Y2 (y1 , y2) = gX1 ,X2 (y1 , y2 − y1 )|J | = 2e−(y1 +y2 −y1 )I(0 < y1 < y2 − y1 )1 = 2e−y2 I(0 < 2y1 < y2).

Y1 ! · · · yn ! The support of Y is Y = {y : n i=1 n i=1 ρyi i . yi ! , n}. The multinomial theorem states that (x1 + · · · + xn )m = Y m! xy11 xy22 · · · xynn . y1 ! · · · yn ! 27) is a pmf. , Yn−1 are important and the nth outcome means that none of the n − 1 important outcomes occurred. , Yik occurred. Then Wk = k−1 k−1 k m − j=1 Yij and P (Wk ) = 1 − j=1 ρij . , ik occurred,” an outcome with probability kj=1 ρij . Hence kj=1 Yij ∼ BIN(m, kj=1 ρij ). Now consider conditional distributions. , Yik .

Notation. The notation for random vectors is rather awkward. In most of the statistical inference literature, Y is a row vector, but in most of the multivariate analysis literature Y is a column vector. In this text, if X and Y are both vectors, a phrase with Y and X T means that Y is a column vector and X T is a row vector where T stands for transpose. Hence in the deﬁnition below, ﬁrst E(Y ) is a p×1 row vector, but in the deﬁnition of Cov(Y ) below, E(Y ) and Y − E(Y ) are p × 1 column vectors and (Y − E(Y ))T is a 1 × p row vector.

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

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