Let 0 0 and 0 Suppose X and Y have joint PDF a Obtain

Let ? > 0, ? > 0 and ? > 0. Suppose X and Y have joint PDF

(a) Obtain the (marginal) PDF of X in terms of c.

(b) Deduce the value of c.

fxy(z, y) = cro-y-1(1-x-y)\"-i, o

Solution

(a)To find marginal density of X integrate joint density with respect to y,

hence marginal density of x i.e. f(x) is

f(x) c dy

        =   c dy

        = c dy

put t = , therefore dy = (1-x) dt

when y =0    and when y = 1-x    t 1

Hence

f(x)   = c dy

         = c dt

which beta integral of first kind with parameters and

f(x)   = c    ( , )      where ( , ) = dt ---------(1)

(b)TO FIND C:

we know that total probability is always equal to one

f(x,y) =     dy ) dx = 1

           = dx    (substituting equation (1) )       =1

         = c ( , ) dx     =1

        = c ( , ) dx     =1 ( by adding and subtracting 1)

    we find again the integrand is an beta integral of first kind with parameters

    therefore,       c   ( , )   ( , ) =1

hence       c =