Page 106 - Applied statistics and probability for engineers

84 Chapter 3/Discrete Random Variables and Probability Distributions

E X i ( ) = 1 p+ ( 0 1 − p) = p
and
2
p −
p p +(0
V X i ( ) = (1 − ) 2 − p) (1 − p) = (1 p)
Sums of random variables are discussed in Chapter 5, and there the intuitively reasonable result that
+
(
E X) = E X ) + E X ) … + E X n )
(
(
(
1
2
is derived. Furthermore, for the independent trials of a binomial experiment, Chapter 5 also
shows that
(
(
V X) = V X ) + V X ) … + V X n )
+
(
(
1
2
p − )
(
Because E X i ) = p and V X i ( ) = (1 p , we obtain the solution E X ( ) = np and V X ( ) =
(
np 1− p).
Mean and
Variance If X is a binomial random variable with parameters p and n,
2
X
np
E
V X
μ = ( ) = np and σ = ( ) = (1 − ) p (3-8)
Example 3-19 Mean and Variance For the number of transmitted bits received in error in Example 3-16, n = 4
and p = 0.1, so
9
.
.
0
1
4
.
0
0
4
E X ( ) = ( ) = 0 4 and V X ( ) = ( )( ) = 0 36
.
1
.
and these results match those obtained from a direct calculation in Example 3-9.
Exercises FOR SECTION 3-6
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
3-91. For each scenario (a)–(j), state whether or not the bino- (g) Reconsider the situation in part (f). Now suppose that the
mial distribution is a reasonable model for the random variable sample of 40 chips consists of chips with 1 and with 0
and why. State any assumptions you make. defects.

(a) A production process produces thousands of temperature trans- (h) A illing operation attempts to ill detergent packages to the

ducers. Let X denote the number of nonconforming transduc- advertised weight. Let X denote the number of detergent
ers in a sample of size 30 selected at random from the process. packages that are underi lled.
(b) From a batch of 50 temperature transducers, a sample of (i) Errors in a digital communication channel occur in bursts that
size 30 is selected without replacement. Let X denote the affect several consecutive bits. Let X denote the number of
number of nonconforming transducers in the sample. bits in error in a transmission of 100,000 bits.
(c) Four identical electronic components are wired to a con- (j) Let X denote the number of surface l aws in a large coil of
troller that can switch from a failed component to one of galvanized steel.
.
the remaining spares. Let X denote the number of compo- 3-92. Let X be a binomial random variable with p = 0 2

nents that have failed after a speciied period of operation. and n = 20. Use the binomial table in Appendix A to determine
(d) Let X denote the number of accidents that occur along the the following probabilities.
(
(
federal highways in Arizona during a one-month period. (a) P X ≤ ) 3 (b) P X > )
10
(
(
(e) Let X denote the number of correct answers by a student (c) P X = ) 6 (d) P 6 ≤ X 11)
≤
taking a multiple-choice exam in which a student can elim- 3-93. Let X be a binomial random variable with p = 0 1.
inate some of the choices as being incorrect in some ques- and n = 10. Calculate the following probabilities from the bino-
tions and all of the incorrect choices in other questions. mial probability mass function and from the binomial table in
(f) Defects occur randomly over the surface of a semiconductor Appendix A and compare results.
chip. However, only 80% of defects can be found by testing. ( (
A sample of 40 chips with one defect each is tested. Let X (a) P X ≤ ) 2 (b) P X > ) 8
(
(
denote the number of chips in which the test i nds a defect. (c) P X = ) 4 (d) P 5 ≤ X ≤ 7)