Page 78 - Probability, Random Variables and Random Processes

70 RANDOM VARIABLES [CHAP 2

2.46. All manufactured devices and machines fail to work sooner or later. Suppose that the failure rate
is constant and the time to failure (in hours) is an exponential r.v. X with parameter A.
Measurements show that the probability that the time to failure for computer memory chips
in a given class exceeds lo4 hours is e- ' (20.368). Calculate the value of the parameter I.
Using the value of the parameter A determined in part (a), calculate the time x, such that
the probability that the time to failure is less than x, is 0.05.
From Eq. (2.49), the cdf of X is given by

Now

from which we obtain 1 =
We want

from which we obtain
x, = - lo4 ln (0.95) = 51 3 hours

2.47. A production line manufactures 1000-ohm (R) resistors that have 10 percent tolerance. Let X
denote the resistance of a resistor. Assuming that X is a normal r.v. with mean 1000 and variance
2500, find the probability that a resistor picked at random will be rejected.
Let A be the event that a resistor is rejected. Then A = {X < 900) u {X > 1100). Since (X < 900) n
{X > 1100) = (21, we have

Since X is a normal r.v. with p = 1000 and a2 = 2500 (a = 50), by Eq. (2.55) and Table A (Appendix A),

Fx(900) = @ (900 ~~ooo) = @( - 2) = 1 - @(2)
F,(1100) = @ ( 5o ) =
1100 - 1000

Thus, P(A) = 2[1 - @(2)] z 0.045

2.48. The radial miss distance [in meters (m)] of the landing point of a parachuting sky diver from the
center of the target area is known to be a Rayleigh r.v. X with parameter a2 = 100.
(a) Find the probability that the sky diver will land within a radius of 10 m from the center of
the target area.
(b) Find the radius r such that the probability that X > r is e- ( x 0.368).
(a) Using Eq. (2.75) of Prob. 2.23, we obtain

(b) Now
P(X > r) = 1 - P(X < r) = 1 - F,(r)
-
- 1 - (1 - e-r2/200) = e-r2/200 = e-l
from which we obtain r2 = 200 and r = $66 = 14.142 rn.