Page 81 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Let y = x2/(2a2). Then dy = x dx/a2, and we get

Next,
I
1 "
= - x2e-x2/(2a2) dx = = . (2.1 08)
2

fi, -"
Then by Eq. (2.31), we obtain

2.53. A r.v. X is said to be without memory, or memoryless, if
P(X~x+tlX>t)=P(Xsx) x,t>O
Show that if X is a nonnegative continuous r.v. which is memoryless, then X must be an expo-
nential r.v.
By Eq. (1.39), the memoryless condition (2.1 10) is equivalent to

If X is a nonnegative continuous r.v., then Eq. (2.1 11) becomes

or [by Eq. (2.2511,
Fx(x + t) - Fx(t) = CFXW - FX(0)ICl - FX(O1
Noting that Fx(0) = 0 and rearranging the above equation, we get

Taking the limit as t -+ 0, we obtain
FW = F>(O)[l - Fx(x)l
where FX(x) denotes the derivative of FX(x). Let
RX(x) = 1 - FX(x)
Then Eq. (2.1 12) becomes

The solution to this differential equation is given by
Rx(x) = keRx(OIx
where k is an integration constant. Noting that k = Rx(0) = 1 and letting RgO) = - FXO) = -fdO) = - 1,
we obtain
Rx(x) = e -lx