Confidence Intervals on Reliability and Availability
When calculating confidence measures on the reliability or availability, the normal approximation of the binomial distribution is used.
Say (1 − α) is the confidence level.
Then:
Where:
p = The probability of interest (reliability or availability).
p L = The lower confidence limit.
p U = The upper confidence limit.
The estimate of p(p) is computed using the number of successful trials (n s) and the total number of trials, where n is the number of simulation runs). The formula is:
Then, lower (p L) and upper (p U) confidence limits should be the solutions of the following equations:
However, for a large n value, these equations can be solved easily using the normal approximation of the binomial distribution, which is used to compute the confidence intervals:
Where z k is the standard normal quantile of the order k:
z k = Φ-1 (k) = -Φ -1(1 − k)
Where Φ(k) is the cumulative distribution function of the standard normal distribution. (Here,
is negative because α/2 < 0.5).
is negative because α/2 < 0.5).The values of p L and p U are checked to make sure that they are in [0,1]. Additionally, the values are set appropriately, such as an appropriate limit (0 or 1).
If the confidence limit is either too low (10-8) or too high (1 − 10-8), then confidence intervals are ignored. The reasoning for this follows:
• If the confidence limit is too low, then both the limits will be the same:
• If the confidence limit is too high, then the limits will be p L = 0 and p U = 10.
Similarly, the limits for
values are checked and results are computed only if 
and the number of simulation runs is greater than 0.
values are checked and results are computed only if and the number of simulation runs is greater than 0.