Page 133 - Fundamentals of Probability and Statistics for Engineers

116 Fundamentals of Probability and Statistics for Engineers

(a) Determine m V and 2 of voltage V, which is given by
V
V ˆ R ‡ r 0 †i:
(b) Determine the correlation coefficient of R and V.
4.23 Let the jpdf of X and Y be given by
xy; for 0 < x < 1; and 0 < y < 2;

f x; y†ˆ
XY 0; and elsewhere:
2 1/2
2
Determine the mean of Z , equal to (X ‡ Y ) .
4.24 The product of two random variables X and Y occurs frequently in applied
problems. Let Z ˆ XY and assume that X and Y are independent. Determine the
2
2
mean and variance of Z in terms of m X , m Y , , and .
X
Y
4.25 Let X ˆ X 1 ‡ X 2 , and Y ˆ X 2 ‡ X 3 . Determine correlation coefficient XY of X
when X 1 , X 2 , and X 3 are uncorrelated.
and Y in terms of X 1 , X 2 , and X 3
4.26 Let X and Y be discrete random variables with joint probability mass function
jpmf) given by Table 4.1. Show that XY ˆ 0 but X and Y are not independent.
Table 4.1 Joint probability mass
function, p XY x, y) for Problem 4.26
y x
1 0 1
1 a b a
0 b 0 b
1 a b a
1
Note: a ‡ b ˆ .
4
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
, 2 ,
storeys X 1 and X 2 . Assume that X 1 and X 2 are independent and let m X 1 , m X 2
X 1
and 2 be their respective means and variances.
X 2
(a) Find the mean and variance of Y .
(b) Find the correlation coefficient between X 2 and Y . Discuss the result if
2 2 .
X 2 X 1
4.28 Let X 1 ,. . ., X n be a set of independent random variables, each of which has a
probability density function (pdf) of the form
1 x =2
2
f x j †ˆ e j ; j ˆ 1; 2; ... ; n; 1 < x j < 1:
X j 1=2
2 †
Determine the mean and variance of Y , where
n
X 2
Y ˆ X :
j
jˆ1

TLFeBOOK