Suppose we are told that and independently.
Find the density function and thus the joint distribution of
To do this, we can obtain the Jacobian and perform a change of variables. Note that it is easier to find the Jacobian of the transformation going for since and are already expressed in terms of and . The Jacobian is given as follows.
The Jacobian is found to be the determinant of above.
Thus, our transformation can be specified as follows. Note that .
Hence, because we see that the pdf factors nicely into a term involving and a term involving , those terms are in fact the marginal densities of and (be sure to make sure that the normalizing constants go to the right places). We recognize the marginal densities as standard normal densities. And thus, it follows that and are i.i.d. .
Let . Find the distribution of in the following two ways.
Find the distribution of using a change of variables
Let . Then we have , and so . Hence, the PDF of is
for . We recognize this PDF as the distribution, and so .
Find the distribution of using a story proof related to the Gamma distribution.
Using the bank-post office story, we can represent with and independent. Then by the same story.
How does and relate to the Binomial distribution?
If we use as the prior distribution for the probability of success in a Binomial problem, interpreting as the number of prior successes and as the number of prior failures, then is the probability of failure and, interchanging the roles of "success" and "failure," it makes sense to have be distributed as the following .
Let be i.i.d. . Find the unconditional distribution of , and the conditional distribution of given .
Unconditionally, , using what we know about Uniform order statistics. For the conditional distribution,
for . Hence, the PDF is . This is the distribution of where , which can be easily verified with a change of variables. Hence, the conditional distribution of is that of a scaled Beta!
Let and , where n is a positive integer and is a positive integer with . Show using a story about order statistics that This shows that the CDF of the continuous r.v. B is closely related to the CDF of the discrete r.v. X, and is another connection between the Beta and Binomial.
Let be i.i.d. . Think of these as Bernoulli trials, where is defined to be "successful" if (so the probability of success is for each trial). Let be the number of successes. Then is the same event as , so .