CORRESPONDENCE BETWEEN POISSON
FAILURE AND NEGATIVE-EXPONENTIAL
LIFE SPANThe failure of a certain type of device has a Poisson probability, and the mean number of
failures in 1500 h of operation is 6. What is the reliability of the device for 270 h of oper-
ation?Calculation Procedure:- Compute the mean life span
Refer to the previous calculation procedure pertaining to the Poisson probability distri-
bution. Consider the following: A device is set in operation. When this device fails, a
device of identical type is set in operation, and this replacement process continues in-
definitely. Let X denote the number of failures in time t, and assume that X has a Pois-
son probability distribution. Thus the probability that exactly 1 failure will occur in 1 h
remains constant as time elapses. It can readily be shown that the life span of the de-
vice, which is the time interval between successive failures, has a negative-exponential
probability distribution.
Select 1 h as the unit of time. Then mean life span IJL = 1500/6 = 250 h. - Compute the reliability
Apply Eq. c of the preceding calculation procedure, giving a = I/JJL= 1/250. Now apply
Eq. b, giving /2(270) = <r270/250 = e~im = 0.340.
PROBABILITY OF FAILURE DURING
A SPECIFIC PERIOD
The life span of a device is negative-exponential, and its mean value is 8 days. What is the
probability that the device will fail on the fifth day?
Calculation Procedure:- Set up the probability equation
Set T = life span, and refer to Fig. 26. As previously stated, P(t{ < T < t 2 ) = area under
curve from J 1 to J 2. It follows that P(Y 1 < T < J 2 ) = RQ 1 ) - R(t 2 ). Take 1 day as the unit of
time. So P(4 < T < 5) - R(4) - R(5). - Compute the probability
Set a =11 IJi= 1/8. Then R(t) = e~at = e~t/s, or R(4) - e^ = 0.607 and R(5) - e~5/ = 0.535.
Thus, P(4 < T < 5) = 0.607 - 0.535 - 0.072.
Related Calculations'. Assume that the device has been operating for some
time. In accordance with the statement previously made, the probability that this device
will fail on the fifth day from the present is 0.072, regardless of the present age of the
device.