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

The wind force W exerted on structures is proportional to V^2 .LetW aV^2.

(a) Determine the pdf of W and its mean and variance by using fW (w).

(b) Determine the mean and variance of W directly from the knowledge of FV (v).

5.11 An electrical device called a full-wave rectifier transforms input X to the device, to
output Y according to Y X. If input X has a pdf of the form

(a) Determine the pdf of Y and its mean and variance using fY (y).

(b) Determine the mean and variance of Y directly from the knowledge of fX (x).

5.12 An electrical device gives output Y in terms of input X according to

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

Suppose that m is deterministic and V is random with pdf given by

Determine the pdf of X.

5.14 The radius R of a sphere is known to be distributed uniformly in the range
0 0 and (b) of its
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

156 Fundamentals of Probability and Statistics for Engineers

FV…v†ˆ

exp

v

36 : 6

 

6 : 96

; forv> 0 ;

0 ; elsewhere:

8

<

:

ˆ

fX…x†ˆ

1

… 2 †^1 =^2

exp



…x 

1 †^2

2

"#

; 

1 <x< 1 ;

Yˆg…X†ˆ

1 ; forX> 0 ;

0 ; forX 0 :



Xˆ

mV^2

2

fV…v†ˆ av

(^2) e bv^2 ; forv> 0 ;
0 ; elsewhere:

: 
fR…r†ˆ
a^2 re^ ar; forr> 0 ;
0 ; elsewhere:

0 99r r1:01r. Determinethepdfs of (a)its surface area
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