Page 68 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

(a) (b)
Fig. 2-19 Rayleigh distribution with o = 1.

(a) Show that the gamma function has the following properties:

(b) Show that the pdf given by Eq. (2.76) satisfies Eq. (2.22).
(c) Plotfx(x) for (a, 1) = (1, I), (2, I), and (5,2).
(a) Integrating Eq. (2.77) by parts (u = xa-', dv = e-" dx), we obtain

Replacing a by a + 1 in Eq. (2.81), we get Eq. (2.78).
Next, by applying Eq. (2.78) repeatedly using an integral value of a, say a = k, we obtain

Since

it follows that T(k + 1) = k! . Finally, by Eq. (2.77),

Let y = x1I2. Then dy = ax- 'I2 dx, and

in view of Eq. (2.73).
(b) Now

Let y = Ax. Then dy = A dx and