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W = x2 which is onetoone transformation as X is nonnegative The inverse is x = w12, and J = dx dw = 1 2 w−1 2 So the pdf of W is g(w) = 2e −2 √ w 1 2 w−1 2 = w 1 2 e−2 √ w, for w > 0 (b) Write down the joint pdf of g(x,y) of X and Y f(x,y) = 2e−2x2e−2y = 4e−2(xy), for x > 0, y > 0 (c) Deﬁne U = X 3Y, and V = Y, then. = p Ï $ Á j U ú X X m M p M M t p K Ð æ å « · Ó ³ ï x ² h å « · Ó ³ ï è Ý 6 t s T î b Ê U V Ï ú. Question 8 Prove Algebraically (a) (w’xy’)(wx’y)(wy’z)=w’x’y’w’yzwxwy’ (c) B’D’AD’C’B’CDBCBA=(DAB’)(CB’)(D’B)(AB.

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Yi(t) = 2 1 x(u)ejtu du Similarly, let y2(t) be the output due to passing x(t) through F twice Y2) = 210 21f x(u)evu du eitv dv = (21r)2 fw Gx(u)fv eftu)vdv du = (2_)2 f 0 x(u)(27r)b(t u) du = (21)' X(t ) Finally, let y(t) be the output due to passing x(t) through F three times yAt = w(t) = 271 (2 7r)x(u)ej'"du = (21r)'f e 'u x(u. è =ù Uù è O Ýù ^Ï è W è p Ï è M ^ D è W è Ý Ù è ý è ¯ ^ ' è ù 7ù èû è M K 7 èä ý ä ^ K Cù è M p Xâ Jù è ù è î @ ù è ý è E $ N è T ^ K 7 è L X D N è P p á ý. Y ¹ t f v a U K Ô ù x z û i w â ¢ ) ò î £ q ¹ w t b V U \ a b { y ¦ Ó ³ ã ï w µ Ö µ § Ì ;.

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