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A stock solution of hydrochloric acid (HCl) supplied by a
Chapter 2, Problem 26SE(choose chapter or problem)
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.