Extra Problems From Ch. 7
1)
Do part (b) of Problem 7.9 on p. 378 of Ross, expressing your final answer as a fairly simple function of n. Assume independence in the placement of the balls.
Provide justification for your answer.
(Note: While part (a) of the problem requires results from Ch. 7, part (b) can be done using results from
just the first three chapters. While doing it will serve as a patial review of the material of first three chapters,
it may also put you in a good frame of mind for doing part (a).
If you have a hard time getting started on this one, you might try
considering all of the ways the balls can be distributed in the urns for the cases of n = 3 and n = 4. It's important
to note that ball i is equally likely to be placed in any of the first i urns ... not equally likely to be placed
in each of the n urns (except for the case of i = n). For example, ball 2 is equally likely to be placed
in urns 1 and 2,
and ball 3 is equally likely to be placed in urns 1, 2, and 3. (Due to this restriction, where must ball n go if each urn is
to have at least one ball?))
2)
Do part (a) of Problem 7.9 on p. 378 of Ross, expressing your answer as a fairly simple function of n. (Don't report your
answer as a messy expression that takes effort to evaluate.) Assume independence in the random placement of the balls.
Provide justification for your answer.
(Note:
If you have a hard time getting started on this one, you might try
considering all of the ways the balls can be distributed in the urns for the cases of n = 3 and n = 4. It's important
to note that ball i is equally likely to be placed in any of the first i urns ... not equally likely to be placed
in each of the n urns (except for the case of i = n). For example, ball 2 is equally likely to be placed
in urns 1 and 2,
and ball 3 is equally likely to be placed in urns 1, 2, and 3. Really, the key to this problem is to note which balls could
possibly go into urn i and to determine the probability that none of them will. (It's not that hard ... things cancel out and
the probability that urn i will be empty (which is a key thing to know if you want the expected number of urns that will be empty)
turns out to be fairly simple.) Also, it's good to recall that the sum of the first n positive integers is n(n+1)/2.)
3)
Do Problem 7.13 on p. 379 of Ross.
Provide justification for your answer.
(Note: To compute the expected value, one doesn't need to know what the ages of the 1000 people are if we assume that each
age is at least 1 and not greater than 1000 (so make this assumption). If we wanted to obtain the variance, we'd need more
information.)
4)
Determine the covariance of the random variables X and Y corresponding to the
joint density given in Problem 6.22 on p. 293 of Ross.
Provide justification for your answer.
(Note:
Obtaining the pdf of X is a problem on HW 9B. You can make use of the answer to that problem if you want to.)