a.
Prove that joint density
a.
Explanation of Solution
Calculation:
In Exercise 5.65 the joint density is given as follows:
Where marginal densities of
It is known that the cumulative density function of
Now, substitute
That is,
Hence, the joint probability density function is obtained as follows:
Hence, it is proved that the joint density function of
b.
Evaluate
b.
Answer to Problem 163SE
The joint cumulative density function of
Explanation of Solution
Calculation:
Consider that
Hence, the
Now, substitute
That is,
Hence, the joint cumulative density function of
c.
Obtain the joint density function associated with the distribution function that is obtained in Part (b).
c.
Answer to Problem 163SE
The joint density function is,
Explanation of Solution
From Part (b), the joint cumulative density function of
Hence, the joint probability density function is obtained as follows:
Thus, the joint density function is,
d.
Provide two specific and different joint densities that yield marginal densities for
d.
Explanation of Solution
Calculation:
From Part (c), the density function is obtained as follows:
Where marginal densities of
The marginal density function does not depend on the values of
Thus, as
Consider
Consider
Hence, these two specific and different joint densities yield marginal densities for
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Chapter 5 Solutions
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