Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 314

The mean value of the Gamma variable is written as:

1 1

l b
GðbÞ

x b e lx d x:

X ¼ xf X ðxÞ d x ¼ (5.87)

0 0

Using integral 1 from Appendix A , we obtain:

l b
GðbÞ

l b
GðbÞ

Gðb þ 1 Þ
l bþ 1

bGðbÞ
l bþ 1 ¼

b
l :

X ¼ ¼ (5.88)

Similarly, the mean squared value is:

l b
GðbÞ

l b
GðbÞ

Gðb þ 2 Þ
l bþ 2

bðb þ 1 ÞGðbÞ
l bþ 2

bðb þ 1 Þ
l 2

X 2 ¼ ¼ ¼ :

(5.89)

Finally, the variance is:

b 2 þ bb 2

l 2

b
l 2 :

s 2

¼ ¼ (5.90)

The Gamma variable is used in queuing theory [MIL04, p. 67].

1

b=0.5

0.9

0.8

0.7

b=1

0.6

b=1.5

0.5

0.4

b=2.5

0.3

0.2

0.1

0 0

1 2 3 4

Range

5 6 7 8

Fig. 5.12 Gamma densities for l ¼ 1 and different values of b