Fundamentals of Probability and Statistics for Engineers

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,
and
d
thei

clearly, Y assumes values 1 and 4, each with probability 1/2, and
their jr joint mass function is given by:

The means and second moment 11 are given by

H ence,

y

x

Figure 4.3 An illustration of perfect correlation,

Expectations and Moments 91

ρ=+1

ρ=–1



 

YˆX^2.

pXY…x;y†ˆ

1

4

; for…x;y†ˆ… 1 ; 1 †;

1

4

; for…x;y†ˆ… 1 ; 1 †;

1

4

; for…x;y†ˆ… 2 ; 4 †;

1

4

; for…x;y†ˆ… 2 ; 4 †:

8

>>

>>

>>

><

>>>

>>

>>

:

mXˆ… 2 †

1

4



‡… 1 †

1

4



‡… 1 †

1

4



‡… 2 †

1

4



ˆ 0 ;

mYˆ… 1 †

1

2



‡… 4 †

1

2



ˆ 2 : 5 ;

11 ˆ… 1 †… 1 †

1

4



‡… 1 †… 1 †

1

4



‡… 2 †… 4 †

1

4



‡… 2 †… 4 †

1

4



ˆ 0 :

11 mXmYˆ 0 ;

Answer: