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Download e-book for iPad: Discrete Multivariate Analysis: Theory and Practice (1977) by Yvonne M. Bishop, Stephen E. Fienberg, Paul W. Holland,

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By Yvonne M. Bishop, Stephen E. Fienberg, Paul W. Holland,

ISBN-10: 0262520400

ISBN-13: 9780262520409

ISBN-10: 0585332045

ISBN-13: 9780585332048

“A great addition to multivariate research. The dialogue is lucid and intensely leisurely, excellently illustrated with purposes drawn from a wide selection of fields. an excellent a part of the publication might be understood with no very really good statistical wisdom. it's a so much welcome contribution to an enticing and energetic subject.” -- Nature initially released in 1974, this e-book is a reprint of a vintage, still-valuable textual content.

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Additional info for Discrete Multivariate Analysis: Theory and Practice (1977)

Example text

The variance of a continuous random variable X is Var X = E (X ; E X ])2 R P Note The properties of expectation and variance are the same for discrete and continuous random variables just replace with in the proofs. Example. Var X = E X 2 ; E X ]2 = Example. 2. TRANSFORMATION OF RANDOM VARIABLES 57 What is the variance of Z ? Var X = E Z 2 ; E Z ]2 Last term is zero Z 1 z2 1 =p z 2e 2 dz 2 ;1 Z 1 z2 1 z2 + e 2 dz = ; p1 ze 2 2 ;1 ;1 =0+1=1 Var X = 1 ; ; ; Variance of X ? 2 Transformation of Random Variables Suppose X1 X2 : : : Xn have joint pdf f (x1 : : : xn ) let Y1 = r1 (X1 X2 : : : Xn ) Y2 = r2 (X1 X2 : : : Xn ) ..

Example (density of products and quotients). Suppose that (X ( f (x y) = 4xy 0 Let U = XY and V = XY for 0 x Otherwise. 2. TRANSFORMATION OF RANDOM VARIABLES 59 r p Y = VU rv y= u r @x = 1 u @v 2 v @y = p1 : @v 2 uv X = UV p x = uv r @x = 1 v @u 2 u @y = ;1 v 12 @u 2 u 32 Therefore jJ j = 21u and so g(u v) = 21u (4xy) r p = 21u 4 uv uv if (u v ) 2 D = 2 uv = 0 Otherwise: Note U and V are NOT independent g(u v) = 2 uv I (u v) 2 D] not product of the two identities. When the transformations are linear things are simpler still.

Since X and Y are jointly continuous random variables P(X 2 A) = P(X 2 A Y 2 (;1 +1)) = where fX (x) = is the pdf of X . Z1 ;1 Z Z1 f (x y)dxdy A ;1 = fA fX (x)dx f (x y)dy Jointly continuous random variables X and Y are Independent if Then P(X f (x y) = fX (x)fY (y) 2 A Y 2 B ) = P(X 2 A) P(Y 2 B ) Similarly jointly continuous random variables X 1 f (x1 : : : xn ) = n Y i=1 : : : Xn are independent if fXi (xi ) 52 CHAPTER 6. CONTINUOUS RANDOM VARIABLES Where fXi (xi ) are the pdf’s of the individual random variables.

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Discrete Multivariate Analysis: Theory and Practice (1977) by Yvonne M. Bishop, Stephen E. Fienberg, Paul W. Holland,

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