144
Chapter 2 Discrete Random Variables and Probability Distributions
8. In exercise 5, suppose now that the entire lot is inspected and any blemished tires
are replaced by good tires if the lot is rejected by the sample. Find the average outgoing quality.
9. In exercise 7, if any defective items in the lot are replaced by good items when the
sample rejects the entire lot, find the average outgoing quality.
10. Exercises 3 and 4 can be generalized. Suppose a box has a red and b blue marbles
and that X is the number of drawings necessary to draw out all of the red marbles.
a. Show that
P. X = x = x − 1
a − 1 a + b
a ;
x = a; a + 1; : : : ; a + b:
b. Using the result in part a, show that a recursion can be simplified to
P. X = x P. X = x − 1
= x − 1
x − a ;
x = a + 1; a + 2; : : : ; a + b:
c. Show that the recursion in part b leads to
a+b
X
x=a+1
x · .x − a · P.X = x =
a+b
X
x=a+1
x · .x − 1 · P.X = x − 1: From this conclude that
E. X = a · a + b + 1
a + 1 :
d. Show that
Var. X = a · b · .a + b + 1
. a + 1
2
· .a + 2 :
11. Exercise 10 continued. Now suppose X represents the number of drawings until
all the marbles remaining in the box are of the same color. Show that P. X = x =
x − 1 a − 1
+ x − 1
b − 1 a + b
a ;
x = Min[a; b]; : : : ; a + b − 1: and that
E. X = a · b
a + 1 +
a · b b + 1
:
12. A box contains 3 red and 5 blue marbles. The marbles are drawn out one at a time
without replacement until a red marble is drawn. Let X denote the total number of drawings necessary.
2.11 Acceptance Sampling Continued
145 a. Find the probability distribution function for X.
b. Find the mean and the variance of X. 13. Exercise 12 is generalized here. Suppose a box contains a red and b blue marbles
and that X denotes the total number of drawings made without replacement until a red marble is drawn.
a. Show that
P. X = x = a + b − x
a − 1 a + b
a , x = 1; 2; : : : ; b + 1:
b. Using the result in part a, show that a recursion can be simplified to
P. X = x P. X = x − 1
= b − x + 2
a + b − x + 1 ;
x = 2; 3; : : : ; b + 1:
c. Use the recursion in part b to show that
E. X = a + b + 1
a + 1 and
Var. X = a · b · .a + b + 1
. a + 1
2
· .a + 2 :
d. Show that the mean and variance in part c approach the mean and variance of
