Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 274

The approximated total time necessary to transmit all N messages is m X + 3 s X .
Xm X þ 3 s X ¼ Nm þ 3 s

p

: (4.212)

Exercise 4.17 A joint probability density of the randomvariables X and Y is given as:

2
8 ð y þ 1 Þ

2

f XY ðx; yÞ ¼ k e ð x 1 Þ

18

: (4.213)

Find the constant k and find the probability

Pf 1 <X< 1 ; 2 <Y< 4 g: (4.214)

Answer From ( 4.213 ), we can see that the random variables X and Y are indepen-
dent normal random variables with the following parameters:

s X ¼ 4 s Y ¼ 9 m X ¼ 1 ; m Y ¼ 1 ; : (4.215)

;

The joint density ( 4.213 ) can be written as:

2 2

e ð x 1 Þ e ð y þ 1 Þ

1
2

1
3

8 18

f XY ðx; yÞ ¼ f X ðxÞf Y ðyÞ ¼ p p

2 p

: (4.216)

2 p

By comparing ( 4.213 ) and ( 4.216 ), we find that k ¼ 1/(12 p ).
Due to the independence of the random variables X and Y , the joint probability
( 4.214 ) can be written as:

Pfð 1 X 1 Þ\ð 2 Y 4 Þg ¼ Pf 1 <X< 1 gPf 2 <Y< 4 g: (4.217)

From ( 4.31 ), we calculate:

erf 1 1

1
2

1 1
2 2

Pf 1 X 1 g ¼ p

2 2

p

erf

¼ 0

1
2 erf

1

¼ p 2 : 3413 ; (4.218)

erf 2 þ 1

1
2

4 þ 1
3 2

Pf 2 Y 4 g ¼ erf p

3 2

p

þ erf

1
2

5
3 2

1
3 2

¼ erf p p ¼ 0 : 5828 : (4.219)

Finally, from ( 4.217 ) to ( 4.219 ), we obtain:

Pfð 1 X 1 Þ\ð 2 Y 4 Þg ¼ Pf 1 X 1 gPf 2 Y 4 g
¼ 0

(4.220)

:

3413 0

:

5828 ¼ 0

: :

1989