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