Fundamentals of Probability and Statistics for Engineers

(a) Determine mV and^2 V of voltage V, which is given by

(b) Determine the correlation co efficient of R and V.

4.23 Let the jpdf of X and Y be given by

Determine the mean of Z, equal to (X^2 Y^2 )1/2.

4.27 In a simple frame structure such as the one shown in Figure 4.7, the total hor-
izontal displacement of top storey Y is the sum of the displacements of individual
storeys X 1 and X 2. Assume that X 1 and X 2 are independent and let mX 1 ,mX 2 ,^2 X 1 ,
and^2 X 2 be their respective means and variances.
(a) Find the mean and variance of Y.
(b) Find the correlation co efficient between X 2 and Y. Discuss the result if

1. 4.28 Let X 1 ,...,Xn be a set of independent random variables, each of which has a
   probability density function (pdf) of the form

Determine the mean and variance of Y, where

116 Fundamentals of Probability and Statistics for Engineers



Vˆ…R‡r 0 †i:

fXY…x;y†ˆ xy; for 0<x<^1 ;and 0<y<^2 ;

0 ; and elsewhere:



‡

4.24 The productof two randomvariablesXandYoccursfrequentlyin applied
problems.LetZˆXYand assumethatXandYare independent.Determinethe
mean and varianceofZin terms ofmX,mY,^2 X, and^2 Y.

4.25 LetXˆX 1 ‡X 2 , andYˆX 2 ‡X 3. DeterminecorrelationcoefficientXYofX
andYin termsofX 1 ,X 2 , andX 3 whenX 1 ,X 2 , andX 3 are uncorrelated.

4.26 LetXandYbe discreterandomvariableswith joint probabilitymass function
jpmf)given by Table4.1. ShowthatXYˆ0 butXandYare not independent.

Table 4.1 Joint probability mass

function,pXYx,y) for Problem 4.26

yx

 101

 1 aba

0 b 0 b

1 aba

Note: a‡bˆ

1

4.





^2 X 2 ^2 X

fXj…xj†ˆ

1

… 2 †^1 =^2

ex

(^2) j= 2
; jˆ 1 ; 2 ;...;n; 1<xj< 1 :
Yˆ
Xn
jˆ 1
Xj^2 :