A candy company distributes boxes of chococlates with a mixture of creams, toffees, and cordials. Suppose that the weight of each box is 1 kg, but the individual weights of the creams, toffees and cordials vary from box to box. For a randomly selected box, let X and Y represent the weights of the creams and the toffees, respectively, and suppose that the joint density function of these var is

f(x,y) = 24xy , 0<= x<=1, 0 <=y<=1, x+y<=1

0 elsewhere

a) find the prob that in a given box the cordials acount for more than 1/2 of the weight.

b) find the mariginal density of the weight of the creams.

c) Find the prob that the weight of the toffees in a box is less than 1/8 of a kg if it is known that creams constitute 3/4 of the weight.

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a) I integrate 24xy from 0 to 1/2 to dy. and my answer= 3/8

however, the correct ans = 1/16

where have i gone wrong ?

b) g(x) = 12xy^2

but the correct answer is 12x(1-x)^2.

why?

c) P(y<1/8 | x=3/4)

= f(x,y) / g(x)

and i got 2 but the correct answer is 1/4.