Question

_{For each of the following functions, determine the value of c for which the function is a joint pdf of two continuous random variables X and Y. (a) f(x,y) = cry, 0 <r<1,7 <y<r. (b) f(r,y) = c(1+r+y), 0 <r<y<1. (c) f( y) = cye.0<r<y.0<y<1. (d) f(x,y) = csin(x+y), 0<I< /2.0 <y < /2. (e) f(x,y) = cr(1-y), 0 <y<1,0 <r<1-y.}

Answer 1

Set (a faigs = cay Osual, x²zyca Since for a valid joint pdf , total probability is equal to 1 ie hoogoo fang) dy du al > SLS* cxy dy dx = 1 f cx (x2-x4) dm = 1 * - 7 = 1 çit = -c=24] Ang © feny) = c (+xy), 0<x<441 con converter Supert of for x=0y=x Sclerat - dx = 1 c[x -+ x - x 7 = 1 tp cing=x O bill (8,0) 0 x ② flaiy) = cyek Osnay?, osyal LS- cyet dy dx = 1 s cet (1-x) dx = 1 > Ç [[-as-ex-cil.ex]? = 1 + q.-2] =! = = Ang @fahy) = C&in (uty , osasta, osya at The C sinchry) ag]on = 1 Sac Ecos(x+y]*%dx = 1 + $52c (cos x +sinx) dx = 1 ac [Binxe- case] The = c.[[1-0)-(0-1)] =1 20=1 [C=% Ang ® flays =Call-ys , osyal, osusly Sisteme cukys ag] du = = $ CXC-30)* dx = L = ['ę x (1 – x4 db = ! → -). -1 = { t = = (= 8 Aas CON