The assumptions underlying a homogeneous Poisson process are sometimes violated.

Constant \(\lambda\)
The rate of events, \(\lambda\) may vary over time or space.
Independence
The occurrence of events may be affected by the occurrence of events in neighbouring times or places.

These two problems often result in a more variable counts than would be expected from a Poisson distribution — called overdispersion.

Location of houses

For a homogeneous Poisson process with rate \(\lambda\) houses per unit area, the MLE for \(\lambda\) is

\[ \hat{\lambda} \;\;=\;\; \overline{X} \;\;=\;\; \frac {911}{1200} \;\;=\;\; 0.7592 \]

The table below shows the sample proportions for each of the counts and the best-fitting Poisson probabilities using \(\hat{\lambda}\) above.

    No of houses,    
\(x\)
Sample
    proportion   
    Poisson probability,   
\(p(x)\)
00.48670.4681
10.33170.3553
20.14000.1349
30.02920.0341
40.00750.0065
50.00330.0010
60.00000.0001
70.00080.0000
80.00000.0000
90.00080.0000
Total 1.0000 1.0000

Zeros and large counts arise more often than expected from a Poisson distribution.

The sample variance is \(S^2 = 0.8902\) which is greater than the sample mean, \(\overline{X} = 0.7592\). Since the mean and variance of a Poisson distribution are equal, this also suggests some overdispersion in the distribution.