A candy company distributes boxes of chocolates with a...

A candy company distributes boxes of chocolates with a mixture of creams, toffees, and cordials. Suppose that the weight of each box is $\frac{1}{2}$ pound, 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 the joint density function of these variables is shown below. Complete parts (a) through (c). $f(x,y) = \begin{cases} 384xy, & 0 \leq x \leq \frac{1}{2}, 0 \leq y \leq \frac{1}{2}, x+y \leq \frac{1}{2} \ 0, & \text{elsewhere} \end{cases}$ (a) Find the probability that in a given box the cordials account for more than $\frac{1}{4}$ of the weight. The probability is $ (Simplify your answer.)

Transcript

00:01 So we have boxes of creams and toffees.

00:05 And let's see here.

00:09 X and y represent the weights of the creams and coffees or creams and toffees in a box.

00:16 So x plus y then must equal the weight of the box, which is one half pound.

00:24 And we are told then that the density for x and y is equal to 384xy for 0 less than or equal to x, less than or equal to 1 half, 0 less than or equal to y, less than or equal to 1 half, and x plus y less than or equal to 1 half.

00:57 Well, that also then, of course, has to be greater than or equal to 0 because x and y are greater than 0, and it is going to be 0 elsewhere.

01:11 So if i draw the area on which this density exists, then let's see here.

01:27 So here is 1, here is 1 half, here is 1, or here is 1, and here is 1 half.

01:49 Then x plus y less than or equal to 1 half.

01:53 Let's see here.

01:55 This is 1 half, this is 1 half.

01:58 So that means then that y is going to be less than or equal to one half minus x so that then is this line here with slope negative one so my density function exists on this area here and now i want to find a probability probability that in a given box the cordials account for more than half the weight.

03:00 So cordials must be greater than one half the weight, which is one half pound.

03:06 So that must be then greater than or equal to one fourth.

03:10 Well, that then is equal to the probability that x plus y, the weight of the creams and toffees, that that must be less than or equal to 1 fourth because those are the same probabilities.

03:27 If the cordials weigh more than 1 fourth, then the creams and toffees make up the other half, so that must be less than half of 1 half, which is 1 fourth...

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