Page 108 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 91

y

ρ =+1

ρ =–1
x

Figure 4.3 An illustration of perfect correlation,

correlation coefficient can vanish when the values of one random variable are
completely determined by the values of another.
Example 4.10. Problem: determine the correlation coefficient of random
variables X and Y when X takes values 1 and 2, each with probability 1/4,
2
and Y ˆ X .
d
Answer: clearly, Y assumes values 1 and 4, each with probability 1/2, and
e
their joint mass function is given by:
th
r
i

j
1
8
> ; for x; y†ˆ 1; 1†;
4
>
>
>
>
> 1
> ; for x; y†ˆ 1; 1†;
< 4
p x; y†ˆ
XY 1
> ; for x; y†ˆ 2; 4†;
>
4
>
>
>
>
> 1
: ; for x; y†ˆ 2; 4†:
4
The means and second moment 11 are given by

1 1 1 1
m X ˆ 2† ‡ 1† ‡ 1† ‡ 2† ˆ 0;
4 4 4 4

1 1
m Y ˆ 1† ‡ 4† ˆ 2:5;
2 2
1 1 1 1

11 ˆ 1† 1† ‡ 1† 1† ‡ 2† 4† ‡ 2† 4† ˆ 0:
4 4 4 4
Hence,
11 m X m Y ˆ 0;
TLFeBOOK