Additional Extra Problems From Ch. 6
1)
Do part (a) of Problem 6.44 on p. 295 of Ross, except just do the case of i = 2.
For this problem, you don't have to show any work.
2)
Do the last portion (i.e., obtain P(X/Y > 1)) of part (c) of Problem 6.44 on p. 295 of Ross.
For this problem, you don't have to show any work.
3)
Do Problem 6.46 on p. 295 of Ross, except be sure to give the density/pdf (instead of the cdf).
Show adequate work to justify your answer.
(Note: You don't have to determine the value of c since it cancels out.)
4)
Do Problem 6.48 on p. 295 of Ross.
Show adequate work to justify your answer.
(Hints: Although you could make use of the joint pdf of three order statistics, I recommend just using the joint pdf
of three iid uniform (0, 1) random variables, and making use of symmetry (e.g., note that
P(X1 > X2 + X3)
= P(X2 > X1 + X3)). I also recommend using the "total probability" result
involving conditioning on a continuous random variable. That is, you can let
Y = X1 + X2, and obtain
P(X3 > Y)
using the fact that
P(X3 > Y | Y = y) =
P(X3 > y) = 1 - y (for 0 < y < 1), and integrating this multiplied by the pdf of Y.
(I.e., you can obtain the unconditional probability that X3 exceeds Y by taking a "weighted average" of
P(X3 > Y | Y = y), doing the averaging by integrating using the density of Y as a
weight function.) For the pdf of the sum of two independent uniform (0,1) random variables, you can use results in the solution of Example 3a
on p. 259 of the text.)
5)
Do Problem 6.51 on p. 295 of Ross, except use the interval (1/3, 2/3) (instead
of (1/4, 3/4)).
Show adequate work to justify your answer.
6)
Do part (b) of Problem 6.52 on p. 295 of Ross, except set λ equal to 1, a equal to 2, and express the
desired probability as a number rounded to the nearest thousandth.
7)
Do part (b) of Problem 6.42 on pp. 294-295 of Ross, except just do the case of i = 4.
For this problem, you don't have to show any work.
8)
Do Problem 6.49 on p. 295 of Ross, except have the motor lifetimes be exponential random variables having mean 1
(i.e., f(x) = exp( - x ), for x > 0).
You don't have to show any work, but be sure to simplify your answer.
9)
Do Problem 6.59 on pp. 295-296 of Ross, except for part (b) just give the pdf of just U (and omit the pdf of V).
Show adequate work to justify your answer.
(Notes: For part (a) be sure to clearly and simply indicate the values of u and v for which the
joint density is nonzero. (Make an accurate sketch if you can't determine a simple way to indicate these values.)
For part (b), you can make use of the joint density obtained in part (a), or you can use the given joint density for
X and Y and use the cdf method.)