Overdispersion
The assumptions underlying a homogeneous Poisson process are sometimes violated.
These two problems often result in a more variable counts than would be expected from a Poisson distribution — called overdispersion.
Location of houses
Assuming a homogeneous Poisson process with rate \(\lambda\) houses per unit area, the maximum likelihood estimate of \(\lambda\) 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, \(p(x)\) |
---|---|---|
0 | 0.4867 | 0.4681 |
1 | 0.3317 | 0.3553 |
2 | 0.1400 | 0.1349 |
3 | 0.0292 | 0.0341 |
4 | 0.0075 | 0.0065 |
5 | 0.0033 | 0.0010 |
6 | 0.0000 | 0.0001 |
7 | 0.0008 | 0.0000 |
8 | 0.0000 | 0.0000 |
9 | 0.0008 | 0.0000 |
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.