Question

Assume that 2 and Z, are two independent random variables that follow the standard normal distribution N(0,1), so that each of them has the density - . - < < . Let X = 22 + 2 Z2, Y = 22 - Z2, S = x2 +Y, and R = xy (e) From (c), please find the densities of X? and Y? (f) From (d) and (e), please find the density of X2 +Y? (=S). (g) From (e), please find the density of (X2Y2) (note that X2 and Y2 are independent from (d)). This is the answer for A,B and C need help with efg 2+7²) -o caco nocy ca flu,y)= - Cucos fin) = tai ēk #193 ] since then, -00 24cm fly)

Answer 1

Hot Zin Non Zn Noon x = 2 2 2 + 2 22 The) EN (0,1) 52 2 - 5 2 22 = 4 ~ No (12) 4 (62)] = N(0) So, xn NCoD and yr Noi e xa Noi) are yoN(0,1)

so, the p.def of x2 is f (x) Kle 22 t OZKO into otherwise yn 2 so, The def of ye is 1/2-1 - 2 if olma o otherwise x? andy? are independent variables./from omd xnx and yhxr X Then, Then, x² + y2 = sn x, ζωχ, Then, additive property of chi-square distribution The p.def of s 9 ( OCS 200 otherwise

f(x) = 8 e/ 2 Foto Otherwise f(*) - 2 0232 0.282 o otherwise We see that, Sn Exponential Distribution with mean 2. x and 42 are independent from lds. Then wecan we can write the joint density of x² and yr as the product of two marginal density of x2 and 42 - ne: (km) . f(K). #y. (m) where x 22 (kum) is the joint density of x and y? 20, CK,m) = fack). fun (m) RV-1 klo 12- 1 2 it LKL owd Olmo OK200 J and (2/2) (172)2 (km)-Y -(Ktm) if olken Ocm200 I and O2m2.00

+x2y2(Kno) (Kum) 2 OLK Loo 2 (E) and olm Loo otherwise