Question

#2

2. Let X, N o ?) for i=1,2. Show that Y = X1 + X, and Z X; - X2 are independent. 3. Let 2-N(0,1) and W x (n) with Z be independent of W. Show that the distribution of T- tudiatvihustion with n deerees of freedom. (Hint: create a second variable U - find the joint distribution

Answer 1

Y+RaRx Y-R = 2 xe hence xert hence x2 = 7-2 By Jacobion bransformation lax32 ay) 2. axe Joint pdf. of (7, 2) a Joint podof. of (x1, A2). 15) KP(Y,Z) = f(A). faly) 151 Ver over xp rolle 12+ (Arrel27 RT02 ale ²1 49182 te - 2012 20 t e 8001242 +222] + 4 foll, e turl. -पश्मिापन

= (p.d,f. of y) (pod, f. 2) - o a podrf of N oll, 202) Ap. def. of a fler, 202) from from o ne sun @ we say that, y fz are independent that y ff are ngeli, 202) Q.3 - Derivation Since t find it's of p. dif. of tn : is a transformation of rt and he pid,f. , : Let , t = 2 Yand Clearly, mo et cos forneo it we get, z = tft f war Hence the Jacobian of transformation is, - ract nyo at 24

17 Joint podof. of (tix) = Joint podof, of (2, w).lol ng (AM) = 4(3, 1NI IT = fire) .farw). lol anm (2 fi cal are independent) T el? V21 2 Nein VER 2 nie - Vatno 220- tcm in To get p. d. f. of t we integrate g (1,7) co.ndy Plt) = g(tiro da - 202

V2710,2") (1+3) Vñ non f1 ++27 m)2 This inle define a todist with ndof, as a conti, r.v. having podof. , f(t) = 1 ſ Vn B (2, 2) (14t2 n 1 olko