3.41 from walpole?
you cannot ignore x+y<1 and you need to draw the region (triangle)
also, a probability cannot exceed 1.
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.
---------------
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.
YOU substitute the X=3/4 and you integrate Y from -infinity to 1/8
note that all these densities are 0 when the variables are negative
so you integrate from 0 to 1/8.
IN GENERAL
Just like a marginal density, you integrate it over the appropriate region.
It's 1 am, I'm tired