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.