Page 193 - Applied statistics and probability for engineers

Section 5-1/Two or More Random Variables 171

and minor defects, respectively. Determine the range of the Determine the following:
(
(
joint probability distribution of X and Y . (a) P X = ) 2 (b) P X = 1 ,Y = ) 2
(
(
5-5. In the transmission of digital information, the probabil- (c) P Z <1 5) (d) P X = 1 or Z = ) 2
.
(
ity that a bit has high, moderate, and low distortion is 0.01, 0.04, (e) E X ( ) (f) P X = | Y = ) 1
1
and 0.95, respectively. Suppose that three bits are transmitted ( (
1
and that the amount of distortion of each bit is assumed to be (g) P X = 1 ,Y = Z | 1 = ) 2 (h) P X = | Y = 1 , Z = ) 2
independent. Let X and Y denote the number of bits with high (i) Conditional probability distribution of X given that Y = 1
and moderate distortion out of the three, respectively. Determine: and Z = 2
(a) f XY ( x, y) (b) f X ( x) 5-9. An engineering statistics class has 40 students; 60% are

(c) E X ( ) (d) f Y| ( y) electrical engineering majors, 10% are industrial engineering
1
majors, and 30% are mechanical engineering majors. A sample
| (
(e) E Y X = ) 1 (f) Are X and Y independent? of four students is selected randomly without replacement for
5-6. A small-business Web site contains 100 pages and 60%, a project team. Let X and Y denote the number of industrial
30%, and 10% of the pages contain low, moderate, and high engineering and mechanical engineering majors, respectively.
graphic content, respectively. A sample of four pages is selected Determine the following:

without replacement, and X and Y denote the number of pages (a) f XY ( x, y) (b) f X ( x) (c) E X ( )
| (
with moderate and high graphics output in the sample. Determine: (d) f Y| ( y) (e) E Y X = ) 3
(a) f XY ( x, y) (b) f X ( x) (f) V Y X| ( 3

= ) 3
(c) E X ( ) (d) f Y| ( y) (g) Are X and Y independent?
3
| (
| (
(e) E Y X = ) 3 (f) V Y X = ) 3 5-10. An article in the Journal of Database Management
(g) Are X and Y independent? [“Experimental Study of a Self-Tuning Algorithm for DBMS
5-7. A manufacturing company employs two devices to Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload
inspect output for quality control purposes. The irst device is used in the TPC-C OLTP (Transaction Processing Performance
able to accurately detect 99.3% of the defective items it receives, Council’s Version C On-Line Transaction Processing) bench-
whereas the second is able to do so in 99.7% of the cases. mark, which simulates a typical order entry application. See
Assume that four defective items are produced and sent out for the following table. The frequency of each type of transaction
inspection. Let X and Y denote the number of items that will (in the second column) can be used as the percentage of each
be identiied as defective by inspecting devices 1 and 2, respec- type of transaction. Let X and Y denote the average number of
tively. Assume that the devices are independent. Determine: selects and updates operations, respectively, required for each
(a) f XY ( x, y) (b) f X ( x) type transaction. Determine the following:

(
(c) E X ( ) (d) f Y| ( y) (a) P X < 5) (b) E X ( )
2
| (
| (
(
| (
(e) E Y X = ) 2 (f) V Y X = ) 2 (c) Conditional probability mass function of X given Y = 0
(d) P X <6 |
Y =
0)
(e) E X Y = ) 0
(g) Are X and Y independent?
5-11. For the Transaction Processing Performance Coun-
5-8. Suppose that the random variables X, Y , and Z have cil’s benchmark in Exercise 5-10, let X, Y , and Z denote the aver-
the following joint probability distribution.
age number of selects, updates, and inserts operations required
, (
x y z f x y z) for each type of transaction, respectively. Calculate the following:
(a) f XYZ (
x, y, z)
,
1 1 1 0.05 (b) Conditional probability mass function for X and Y given Z = 0
(
| (
6
1 1 2 0.10 (c) P X < ,Y <6 | Z = 0) (d) E X Y = 0 ,Z = ) 0
1 2 1 0.15 5-12. In the transmission of digital information, the prob-
1 2 2 0.20 ability that a bit has high, moderate, or low distortion is 0.01,
0.04, and 0.95, respectively. Suppose that three bits are trans-
2 1 1 0.20
mitted and that the amount of distortion of each bit is assumed
2 1 2 0.15
to be independent. Let X and Y denote the number of bits with
2 2 1 0.10
high and moderate distortion of the three transmitted, respec-
2 2 2 0.05
tively. Determine the following:
Average Frequencies and Operations in TPC-C
Transaction Frequency Selects Updates Inserts Deletes Non-unique Selects Joins
New order 43 23.0 11 12 0 0 0
Payment 44 4.2 3 1 0 0.6 0
Order status 4 11.4 0 0 0 0.6 0
Delivery 5 130.0 120 0 10 0 0
Stock level 4 0 0 0 0 0 1