Page 173 - Fundamentals of Probability and Statistics for Engineers

156 Fundamentals of Probability and Statistics for Engineers

5.10 At a given location, the PDF of annual wind speed, V, in miles per hour is found
to be

8
v
6:96
exp
; for v > 0;
<
F V v†ˆ 36:6
:
0; elsewhere:
2
2
The wind force W exerted on structures is proportional to V . Let W ˆ aV .
(a) Determine the pdf of W and its mean and variance by using f W (w).
(b) Determine the mean and variance of W directly from the knowledge of F V (v).
5.11 An electrical device called a full-wave rectifier transforms input X to the device, to
output Y according to Y ˆj j
X . If input X has a pdf of the form
" 2 #
1
x
1†
f x†ˆ 1=2 exp ;
1 < x < 1;
X
2 † 2
(a) Determine the pdf of Y and its mean and variance using f (y).
Y
(b) Determine the mean and variance of Y directly from the knowledge of f (x).
X
5.12 An electrical device gives output Y in terms of input X according to
1; for X > 0;

Y ˆ g X†ˆ
0; for X 0:
Is random variable Y continuous or discrete? Determine its probability distribution
in terms of the pdf of X.
5.13 The kinetic energy of a particle with mass m and velocity V is given by
mV 2
X ˆ
2
Suppose that m is deterministic and V is random with pdf given by

2
2
bv
av e ; for v > 0;
f v†ˆ
V
0; elsewhere:
Determine the pdf of X.
5.14 The radius R of a sphere is known to be distributed uniformly in the range
:
: 0 99 r 0 1 01 r . Determine the pdfs of (a) its surface area and (b) of its
r
0
volume.
5.15 A resistor to be used as a component in a simple electrical circuit is randomly
chosen from a stock for which resistance R has the pdf
2
ar
a re ; for r > 0;
f r†ˆ
R
0; elsewhere:

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