We will now use a pivot to find a confidence interval for the rate parameter, \(\lambda\), of a homogeneous Poisson process, based on a random sample of inter-event times that have \(\ExponDistn(\lambda)\) distributions.
Example
The following table shows the number of operating hours between successive failures of air-conditioning equipment on one aircraft.
487 | 18 | 100 | 7 | 98 | 5 | 85 | 91 | 43 | 230 | 3 | 130 |
Assuming that failures arise at random over time with a constant rate, \(\lambda\) — that failures are a homogeneous Poisson process — the values are a random sample from an \(\ExponDistn(\lambda)\) distribution. Find a 95% confidence interval for the failure rate, \(\lambda\) per hour.
We showed earlier that the maximum likelihood estimator of \(\lambda\) based on a random sample from an \(\ExponDistn(\lambda)\) distribution is the inverse of the sample mean,
\[ \hat{\lambda} \;\;=\;\; \frac n {\sum{X_i}} \]We will now find a pivot based on the sum of the sample values. We showed earlier that the sum of \(n\) independent exponential variables has an Erlang distribution — a Gamma distribution whose first parameter is integer,
\[ \sum_{i=1}^n{X_i} \;\;\sim\;\; \ErlangDistn(n, \lambda) \;=\; \GammaDistn(n, \lambda) \]This is not a pivot since it depends on \(\lambda\), but the inverse of Gamma distribution's second parameter is a scale parameter, so
\[ \lambda \sum_{i=1}^n{X_i} \;\;\sim\;\; \GammaDistn(n, 1) \]and this is a pivot.
Since \(n = 12\) for this data set, we can use Excel to find the 2½% and 97½% points of the \(\GammaDistn(12, 1)\) distribution — they are 6.201 and 19.682. A 95% confidence interval is therefore the values of \(\lambda\) for which
\[ 6.201 \;\;\lt\;\; \lambda \sum_{i=1}^n{x_i} \;\;\lt\;\; 19.682 \] \[ 6.201 \;\;\lt\;\; 1297\lambda \;\;\lt\;\; 19.682 \]Rearranging these inequalities gives
\[ 0.0048 \;\;\lt\;\; \lambda \;\;\lt\;\; 0.0152 \]We are therefore 95% confident that the failure rate of the air-conditioning unit on this plane is between 4.8 and 15.2 times per thousand hours.