Concept explainers
A stock solution of hydrochloric acid (HC1) 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
- 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.
a.
Find the value of the constant c.
Answer to Problem 26SE
The value of the constant c is
Explanation of Solution
Given info:
The joint probability density function of X and Y is,
Calculation:
The total probability of the joint density function is 1.
That is,
The value of the constant is obtained as follows:
Thus, the value of the constant c is
b.
Compute the marginal density function
Answer to Problem 26SE
The marginal density function
Explanation of Solution
Calculation:
From part (a), value of the constant c is
Substitute value of the constant c in joint probability density function of X and Y is,
The formula for finding marginal probability density function
The marginal probability density function
For
Here,
Therefore,
For
Here,
Therefore,
Thus, the marginal density function
c.
Compute the conditional density function
Answer to Problem 26SE
The conditional density function
Explanation of Solution
Observation:
The formula for finding the conditional density function is,
For
Here,
For
If
Therefore,
If
Thus, the conditional density function
d.
Compute the conditional expectation
Answer to Problem 26SE
The conditional expectation
Explanation of Solution
Calculation:
The value of x is 0.4.
Substitute 0.4 in
The conditional expectation
Thus, the conditional expectation
e.
Explain whether X and Y are independent or not.
Answer to Problem 26SE
Yes, X and Y are not independent because
Explanation of Solution
Calculation:
If X and Y are independent then
From part (c), the conditional density function
For
The formula for finding marginal probability density function
The marginal probability density function
For
Here,
Therefore,
For
Here,
Therefore,
Now
Thus, the X and Y are not independent.
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Chapter 2 Solutions
Statistics for Engineers and Scientists
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