The following symbols and notations are common to test methods 1 - 3 contained in this appendix:
X = the random variable which denotes the maintenance
characteristics of interest (e.g., X can denote corrective maintenance time,
preventive maintenance time, fault location time, manhours per maintenance
Xi = the ith; observation or value of the random variable X.
n = the sample size.
X = the sample mean
E(random variable) = the expected value of the variable
2 = E[(lnX - )2] = the true variance of lnX.
µ = E(X) = the true mean of X.
2 = Var (X) = E[( X - µ2] = the true variance of X.
2 = the sample variance of X (i.e., d2 =
2 = the prior estimate of the variance of the maintenance time.
Xp = the (1-p)th percentile of X (i.e., X.50 = the 95th percentile of X).
= X.50 = the median of X.
Y = lnX = the natural logarithm of X.
= the sample mean of Y
= E(lnX) = the true mean of lnX
2 = the prior estimate of the variance of the logarithm of maintenance times.
s2 = the sample variance of lnX.
Zp = the standardized normal deviate exceeded with probability p (i.e.,
Z = Z(1 - ) =
standardized normal deviate exceeded with probabilities and (1 - ) respectively.
= the producer's risk; the probability that the equipment will be rejected when it has a true value equal to the desired value (H0).
= the consumer's risk; the probability that the equipment will be accepted when it has a true value equal to the maximum tolerable value (H1).
H0 = the desired value specified in the contract or specification and is expressed as a mean, critical percentile, or critical maintenance time.
H1 = the maximum tolerable value. Note: H0 < H1.
When X is a log-normally distributed random variable:
If Y = lnX, the probability density of Y is normal with mean and 2 variance
Y ~ N( , 2 )
Properties of the log-normal distribution:
mean = µ = e
|variance = d2 = e
median = = e
mode = M = e
(1-p)th percentile = Xp = e
Table B-VIII - Standardized Normal
The following symbols are common to test methods 4, 8 - 11 contained in this appendix.
Xci = Maintenance downtime per corrective maintenance task (of the ith task).
Xpmi = Maintenance downtime per preventive maintenance task (of the ith task).
nc = Number of corrective maintenance tasks sampled.
npm = Number of preventive maintenance tasks sampled.
= That value, corresponding to risk, which is obtained from a table of normal distribution for a one-tail test.
fc = Number of expected corrective maintenance tasks occurring during a representative operating time (T).
fpm = Number of expected preventive maintenance tasks occurring during a representative operating time (T).
T = Item representative operating time period.
Dt = Total maintenance downtime in the representative operating time (T).
c, pm, p/c = Mean downtimes of sample. (Corrective, Preventive, and combined Corrective/Preventive Maintenance Times.)
M'Maxc = Sample calculated maximum corrective maintenance downtime.
µc = Specified mean corrective maintenance time.
µpm = Specified mean preventive maintenance time.
µp/c = Specified mean maintenance time. (Taking both corrective and preventive maintenance time into account.)
MMax = A requirement levied in terms of a maximum value of a percentile of task time (i.e., 95% of all corrective task times must be less than 60 minutes) usually taken as the 90th or 95th percentile.
MMaxc = Specified MMax of corrective maintenance downtimes.
MMaxpm = Specified MMax of preventive maintenance downtimes.
c = E(ln Xc ) = Expected value of the logarithm of corrective maintenance tasks.
Log Xci, Log Xc =
Log to the base 10 of Xci, Xc.
ln Xci, ln Xc =
Natural logs of Xci, Xc.
Median value of corrective maintenance tasks.
Median value of preventive maintenance tasks.