Problem 26SE

A stock solution of hydrochloric acid (HCl) supplied by a certain vendor contains small amounts of several impurities, including copper and nickel. Let X denote the amount of copper and let Y denote the amount of nickel, in parts per ten million, in a randomly selected bottle of solution. Assume that the joint probability density function of X and Y is given by

a. Find the value of the constant c so that f(x, y) is a joint density function.

b. Compute the marginal density function fX(x).

c. Compute the conditional density function fY|X(y|x).

d. Compute the conditional expectation E(Y|X = 0.4).

e. Are X and Y independent? Explain.

Solution :

Step 1 of 5

Given, the amount of copper is X and the amount of nickel is Y.

We assume that X and Y are the joint probability density function.

Our goal is to find :

a). Find the value of the constant c so that f(x,y) is a joint density function.

b). Compute the marginal density function .

c). Compute the conditional density function .

d). Compute the conditional expectation of .

e). Are X and Y are independent.We need to explain why?

a).

Now we have to find the value of the constant c so that f(x,y) is a joint density function.

Now we are finding c value.

So

Then,

We integrated then we get.

Therefore constant c value is 0.8571.