A stock solution of hydrochloric acid (HCl) supplied by a

Chapter 2, Problem 26SE

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QUESTION:

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  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  and  is given by

\(f(x, y)= \begin{cases}c(x+y)^{2} & 0<x<1 \text { and } 0<y<1 \\ 0 & \text { otherwise }\end{cases}\)

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

b. Compute the marginal density function \(f_X(x)\).

c. Compute the conditional density function \(f_{Y \mid X}(y \mid x)\).

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

e. Are  and  independent? Explain.

Equation Transcription:

Text Transcription:

(HCl)

f(x,y)={_0 otherwise ^c(x+y)^2 0<x<1 and 0<y<1

f(x,y)

f_X(x)

f_Y|X(y|x)

E(Y|X=0.4)

Questions & Answers

QUESTION:

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  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  and  is given by

\(f(x, y)= \begin{cases}c(x+y)^{2} & 0<x<1 \text { and } 0<y<1 \\ 0 & \text { otherwise }\end{cases}\)

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

b. Compute the marginal density function \(f_X(x)\).

c. Compute the conditional density function \(f_{Y \mid X}(y \mid x)\).

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

e. Are  and  independent? Explain.

Equation Transcription:

Text Transcription:

(HCl)

f(x,y)={_0 otherwise ^c(x+y)^2 0<x<1 and 0<y<1

f(x,y)

f_X(x)

f_Y|X(y|x)

E(Y|X=0.4)

ANSWER:

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.


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