Extra Problems From Ch. 6
1)
Do part (a) of Problem 6.2 on p. 291 of Ross, except change the number of white balls from 5 to 4.
Express the desired joint pmf in the form I use in the example on p. 6-3 of the course notes.
For this problem, you don't have to show any work.
2)
If two points are randomly selected on a line segment of length L so that they are on opposite sides of the midpoint of the line segment,
what is the probability that the distance between the two points is greater than L/4.
(That is, if X is uniformly distributed on the interval (0, L/2) and Y
is uniformly distributed on the interval (L/2, L), if X and Y are independent, what is the value of P(Y - X > L/4)?
Show adequate work to justify your answer.
(Hint: Make a careful sketch showing both where the joint density is positive and also the (x, y) points
satisfying the event under consideration. (See p. 6-9 of the course notes for the type of sketches I'm referring to.)
Once you make the sketch, you can obtain the desired probability using integration, or geometry.)
3)
Do part (a) of Problem 6.19 on p. 293 of Ross.
Show adequate work to justify your answer.
(As a check of your method, if you do part (b) you should find that X has a uniform (0, 1) distribution.)
4)
Do the second part (i.e., obtain the stated probability) of Problem 6.27 on p. 293 of Ross, except
let λ1 = 1 and
let λ2 = 2.
Show adequate work to justify your answer.
5)
Do part (a) of Problem 6.29 on p. 293 of Ross, except obtain the probability that the total gross sales over the next two weeks
are less than $4000. Assume the sales for each of the two weeks are independent random variables having the distribution indicated.
Show adequate work to justify your answer.
6)
Do Problem 6.14 on p. 292 of Ross, reporting your answer by giving a cdf.
Show adequate work to justify your answer.
(Hint: Make a careful sketch showing both where the joint density is positive and also the (x, y) points
satisfying the event under consideration. (Recall that |d| < c if and only if -c < d < c.)
Once you make the sketch, you can obtain the desired probability using integration, or geometry.)
7)
Do part (d) of Problem 6.19 on p. 293 of Ross.
Show adequate work to justify your answer.
(As a check of your method, if you do part (c) you should find that E(X) = 1/2.)
8)
Do part (b) of Problem 6.22 on p. 293 of Ross.
Show adequate work to justify your answer.
9)
Do part (c) of Problem 6.22 on p. 293 of Ross, except obtain the probability that the sum is less than 1/2 (instead of less than 1).
Show adequate work to justify your answer.
10)
Do the first part (i.e., obtain the distribution of the ratio) of Problem 6.27 on p. 293 of Ross, except
let λ1 = 1 and
let λ2 = 2.
Report your answer by giving a pdf (a density function).
Show adequate work to justify your answer.