Page 89 - Fundamentals of Probability and Statistics for Engineers

72 Fundamentals of Probability and Statistics for Engineers

9c) P9X 0:5jY ˆ 1:5).
9d) P9X 0:5jY 1:5).
3.17 Let random variable X denote the time of failure in years of a system for which the
PDF is F X (x). In terms of F X (x), determine the probability
P X xjX 100†;

which is the conditional distribution function of X given that the system did not fail
up to 100 years.
3.18 The pdf of random variable X is

3x ; for 1 < x 0;
2
f x†ˆ
X
0; elsewhere:

j
Determine P(X > b X < b/2) with 1 < b < 0.
3.19 Using the joint probability distribution given in Example 3.5 for random variables
X and Y , determine:
(a) P(X > 3).
(b) P(0 Y < 3).
(c) P(X > 3 Y 2).
j
3.20 Let
x‡y†
ke ; for 0 < x < 1; and 0 < y < 2;
f XY x; y†ˆ
0; elsewhere:
(a) What must be the value of k?
(b) Determine the marginal pdfs of X and Y .
(c) Are X and Y statistically independent? Why?
3.21 A commuter is accustomed to leaving home between 7:30 a.m and 8:00 a.m., the drive
to the station taking between 20 and 30 minutes. It is assumed that departure time and
travel time for the trip are independent random variables, uniformly distributed over
their respective intervals. There are two trains the commuter can take; the first leaves
at 8:05 a.m. and takes 30 minutes for the trip, and the second leaves at 8:25 a.m. and
takes 35 minutes. What is the probability that the commuter misses both trains?
3.22 The distance X (in miles) from a nuclear plant to the epicenter of potential earth-
quakes within 50 miles is distributed according to
2x
8
; for 0 x 50;
<
f x†ˆ 2500
X
0; elsewhere;
:
and the magnitude Y of potential earthquakes of scales 5 to 9 is distributed
according to
8 2
3 9 y†
<
f y†ˆ 64 ; for 5 y 9;
Y
0; elsewhere:
:

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