Calculation Procedure:
- Compute the probability that a particular satellite will survive
the first year
Refer to the earlier calculation procedure pertaining to the negative-exponential probabil-
ity distribution. A device has a negative-exponential life span if its life-span curve has an
equation of this form:/(O = ae~at, Eq. a, where a = positive constant and e - base of natu-
ral logarithms. From Eq. 16a, R(t) = e~at, Eq. b. Figure 29 presents the life-span and relia-
bility diagrams. The mean life span JJL = I/a, Eq. c.
Take 1 month as the unit of time. In this case, JJL = 15; then a = 1/15. By Eq. b, R(t) =
e~t/l5, or /£(12) = eu/l5 = er°* = l/e°-s = 0.4493. Thus, there is a probability of 0.4493 that
a particular satellite will survive the first year. - Establish the probability distribution of X
Refer to the earlier calculation procedure pertaining to the binomial probability distribu-
tion. Launching a satellite may be viewed as a trial to determine whether the satellite will
survive the first year. Since all satellites operate independently, the trials are independent
of one another, and therefore X has a binomial probability distribution.
Apply the equation in step 2 of the procedure for the binomial distribution, with n = 4,
P = 0.4493, and 1 -P = 0.5507. Now, C 40 = C 44 = 1; C 41 = C 43 = 4; C 42 - 6. ThenP(O)
= 1(0.5507)^4 = 0.092; P(I) = 4(0.4493)(6.5507)^3 = 0.300; P(2) = 6(0.4493)^2 (0.5507)^2 =
0.367; P(3) = 4(0.4493)^3 (0.5507) - 0.200; P(4) = !(0.4493)^4 = 0.041. These probabilities
total 1, as they must.
Related Calculations: It can readily be shown that a device that has a negative-
exponential life span and has been operating for some time has the same probability of
surviving the next unit of time as one that was just activated. Thus, the age of the device is
completely irrelevant in predicting its remaining life. Thus failure is caused not by cumu-
lative damage resulting from use but by a sudden accidental occurrence, and the probabil-
ity of this occurrence during the next unit of time is independent of the age of the device.
To apply an analogy, the probability that an individual who travels extensively by plane
will become the victim of a plane crash during the next year has no relation to the amount
of air travel that person has done in the past.
FIGURE 29Life-span curve Reliability curve