Example 4.18. Let X and Y be i.i.d. - N (0, o²), so that fxr(x, y) = fx(x)fy(y) 1 1 1 for all æ and y. Let Z = X² + Y². Then fz(z)dz fxy(x, y)dædy {(1,y):/a VI²+y²€[z,z+dz)} 27 z+dz fxy(r cos 0, r sin...

Work out Example 4.18 in the lecture notes by using the CDF method.

Extracted text: Example 4.18. Let X and Y be i.i.d. - N (0, o²), so that fxr(x, y) = fx(x)fy(y) 1 1 1 for all æ and y. Let Z = X² + Y². Then fz(z)dz fxy(x, y)dædy {(1,y):/a VI²+y²€[z,z+dz)} 27 z+dz fxy(r cos 0, r sin 0)rdrd® (cf. Figure 4.8) -2T z+dz „² cos² 0+r² sin² , ´rdrd0 e 1 r2m z+dz rdrd0 27o? -2T 1 zdzd0 2no? -27 1 do 2no2 zpz - 1 e¯20* zdz(27) * zpZ - Cancelling dz on both sides, we have 1 fz(z) = for all z > 0. Obviously, fz(z) = 0 for all z < 0.="" the="" reader="" is="" encouraged="" to="" repeat="" the="" above="" example="" by="" the="" cdf="">