Page 54 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

Fig. 2-7 Uniform distribution over (a, b).

A uniform r.v. X is often used where we have no prior knowledge of the actual pdf and all
continuous values in some range seem equally likely (Prob. 2.69).

E. Exponential Distribution:

A r.v. X is called an exponential r.v. with parameter A (>O) if its pdf is given by

which is sketched in Fig. 2-8(a). The corresponding cdf of X is

which is sketched in Fig. 2-8(b).

Fig. 2-8 Exponential distribution.

The mean and variance of the exponential r.v. X are (Prob. 2.32)

The most interesting property of the exponential distribution is its "memoryless" property. By
this we mean that if the lifetime of an item is exponentially distributed, then an item which has been
in use for some hours is as good as a new item with regard to the amount of time remaining until the
item fails. The exponential distribution is the only distribution which possesses this property (Prob.
2.53).