Page 91 - Applied statistics and probability for engineers

Section 3-2/Probability Distributions and Probability Mass Functions 69

(
Consider a few special cases. We have P X = 1) = P p ( ) = .
0 01. Also, using the independence assumption,
P X = ) = ( . (0 01. ) = 0 0099.
(
P ap) = 0 99
2
A general formula is
P X = ) P aa … ) x−1 (0.01 ), for x = 1, 2, 3,…
x = (
(
,
ap = 0.99

( x −1) a’s
Describing the probabilities associated with X in terms of this formula is a simple method to dei ne the distribution of X
in this example. Clearly f x ( ) ≥ 0. The fact that the sum of the probabilities is 1 is left as an exercise. This is an example
of a geometric random variable for which details are provided later in this chapter.
Practical Interpretation: The random experiment here has an unbounded number of outcomes, but it can still be
conveniently modeled with a discrete random variable with a (countably) ini nite range.
Exercises FOR SECTION 3-2

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

3-16. The sample space of a random experiment is {a, b, c, d, 3-21. x 1.25 1.5 1.75 2 2.25
e, f}, and each outcome is equally likely. A random variable is f x ( )
.
.
.
.
.
deined as follows: 0 2 0 4 0 1 0 2 0 1

(
.
outcome a b c d e f (a) P X( ≥ 2 ) (b) P X < 165 )
(
(c) P X = 1. ) 5 (d) P X < 1 .3 or X > 21 )
(
x 0 0 1.5 1.5 2 3
3-22. Consider the hospital patients in Example 2-8. Two
Determine the probability mass function of a. Use the patients are selected randomly, with replacement, from the
probability mass function to determine the following total patients at Hospital 1. What is the probability mass func-
probabilities: tion of the number of patients in the sample who are admitted?
. (
(
(a) P X = ) 5. (b) P 0 5 < X < 2 7) 3-23. An article in Knee Surgery, Sports Traumatology,
.
1
Arthroscopy
[“Arthroscopic Meniscal Repair with an Absorb-
(
(
(c) P X > ) 3 (d) P 0 ≤ X < 2) able Screw: Results and Surgical Technique” (2005, Vol. 13, pp.
(
(e) P X = 0 or X = ) 2 273–279)] cites a success rate of more than 90% for meniscal tears
with a rim width under 3 mm, but only a 67% success rate for tears
For Exercises 3-17 to 3-21, verify that the following functions
are probability mass functions, and determine the requested of 3–6 mm. If you are unlucky enough to suffer a meniscal tear of
probabilities. under 3 mm on your left knee and one of width 3–6 mm on your
3-17. x –2 –1 0 1 2 right knee, what is the probability mass function of the number of
successful surgeries? Assume that the surgeries are independent.
.
.
.
f x ( ) 0 2. 0 4 0 1 0 2 0 1 3-24. An optical inspection system is used to distinguish
.
among different part types. The probability of a correct classii -
(
(
(a) P X ≤ ) 2 (b) P X > − ) 2 cation of any part is 0.98. Suppose that three parts are inspected
(
(
(c) P − Ð1 X≤ ) 1 (d) P X ≤ −1 or X = ) 2 and that the classii cations are independent. Let the random
variable X denote the number of parts that are correctly classi-
x
=
(
/
/
3-18. f x) ( 8 7 )(1 2 ) , x = 1 , ,3 ied. Determine the probability mass function of X.
2

(a) P X( ≤1 ) (b) P X( >1 ) 3-25. In a semiconductor manufacturing process, three
(c) P(2 < X < ) 6 (d) P X( ≤ 1 or X > 1 ) wafers from a lot are tested. Each wafer is classii ed as pass or
fail. Assume that the probability that a wafer passes the test is 0.8
x +1
2
(
3-19. f x) = , x = 0 1 2 3 4 and that wafers are independent. Determine the probability mass
, , , ,
25
(
(
(a) P X = 4 ) (b) P X ≤ 1 ) function of the number of wafers from a lot that pass the test.
(c) P(2 ≤ X < ) 4 (d) P X( > −10 ) 3-26. The space shuttle l ight control system called Pri-
mary Avionics Software Set (PASS) uses four independent
3-20. f x( ) ( / )( / )= 3 4 1 4 x , x = 0 , , ,… computers working in parallel. At each critical step, the com-
1
2
(
(
(a) P X = 2) (b) P X ≤ ) 2 puters “vote” to determine the appropriate step. The probability
that a computer will ask for a roll to the left when a roll to
(
(
(c) P X > ) 2 (d) P X ≥ ) 1 the right is appropriate is 0.0001. Let X denote the number of