Question

Let X1 and X2 be two continuous random variables with joint PDF fX1,X2. Let Y1 = X1 + X2 and Y2 = X1 - X2. (a) Find the joint PDF of Y1 and Y2 in terms of fX1,X2. Suppose, in addition, that X1 and X2 are independent standard normals. (b) Show that Y1 and Y2 are independent and find their PDFs fY1 and fY2. 

          Let X1 and X2 be two continuous random variables with joint PDF fX1,X2. Let Y1 = X1 + X2 and Y2 = X1 - X2.

(a) Find the joint PDF of Y1 and Y2 in terms of fX1,X2.
Suppose, in addition, that X1 and X2 are independent standard normals.

(b) Show that Y1 and Y2 are independent and find their PDFs fY1 and fY2.
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Added by Joaquin W.

Elementary Statistics a Step by Step Approach
Elementary Statistics a Step by Step Approach
Allan G. Bluman 9th Edition
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Let X1 and X2 be two continuous random variables with joint PDF fX1,X2. Let Y1 = X1 + X2 and Y2 = X1 - X2. (a) Find the joint PDF of Y1 and Y2 in terms of fX1,X2. Suppose, in addition, that X1 and X2 are independent standard normals. (b) Show that Y1 and Y2 are independent and find their PDFs fY1 and fY2.
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Transcript

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00:01 Here the joint probability density function of y1 comma y2 in terms of x1 x2 is given as fy 1 comma y2 of y 1 y2 equal to f x1 x2 of y 1 plus y2 by 2 comma y 1 minus y2 by 2 multiplied by x so which is equal to half times f x1 x1 plus y2 by 2 by 2.
00:30 Comma y1 minus y2 by 2 which is equal to 1 by root of 2 pi e power minus x1 square by 2 multiplied by 1 by root of 2 p e power minus x2 square by 2 then next part is f y 1 1 by 2 of y 1 y2 equal to 1 by 2 5 e power minus 1 by 2 times y 1 plus y 2 by 2 by 2 the whole square plus y1 minus y2 by 2 the whole square, which is equal to 1 by 4 pi e power minus 1 by 2 times y 1 square by 2 plus y2 square by 2 which is equal to 1 by root 2 ,0 of 2 pi, e power minus 1 by 4 y1 square multiplied by 1 by root 2, 0, e power minus y2 square by 4, where y 1 1 .1, 3 ,000, 2, 0 ,000, 2, e power minus y2 square by 4, where y 1 1, belongs to r and y2 belongs to r...
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