Fitting Poisson

16.5 Fitting a Poisson Distribution

Fitting a Poisson distribution involves assessing how well a Poisson model represents actual observed data. This process is analogous to finding the best-fitting statistical model to describe the frequency of rare events occurring over a fixed interval of time or space.

For instance, consider analyzing the frequency of equipment failures in a manufacturing facility. If these failures occur independently and at a relatively low rate, the Poisson distribution can serve as a suitable model.

To fit the distribution, we compare the observed frequencies of events with the theoretical frequencies predicted by the Poisson probability formula. A close agreement between observed and theoretical frequencies indicates that the Poisson model is appropriate for the data, allowing for reliable conclusions and predictions.

Poisson Distribution Formula

The Poisson probability mass function (PMF) is given by:

$$ P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!} $$

Where:

• $P(X = x)$: The probability of observing exactly $x$ events.

• $\lambda$ (lambda): The average rate or mean number of occurrences in a given interval.

• $e$: The base of the natural logarithm, approximately 2.71828.

• $x$: The number of events occurring, where $x \in {0, 1, 2, \dots}$.

Example: System Failures in a Factory

Let's consider an example where a sample of 200 machines in a factory was monitored for system errors during one week. The number of machines experiencing 0, 1, 2, 3, 4, or 5 system failures was recorded.

| No. of System Failures ($x$) | No. of Machines ($f$) | | :------------------------- | :-------------------- | | 0 | 80 | | 1 | 72 | | 2 | 38 | | 3 | 6 | | 4 | 3 | | 5 | 1 | | Total | 200 |

Step-by-Step Solution for Fitting

Step 1: Calculate the Mean ($\lambda$)

The mean ($\lambda$) of the observed data is calculated by finding the sum of the products of each number of failures ($x$) and its corresponding frequency ($f$), divided by the total number of machines.

• Sum of $fx$: $(0 \times 80) + (1 \times 72) + (2 \times 38) + (3 \times 6) + (4 \times 3) + (5 \times 1)$ $= 0 + 72 + 76 + 18 + 12 + 5 = 183$

• Mean ($\lambda$): $\lambda = \frac{\text{Total } fx}{\text{Total number of machines}} = \frac{183}{200} = 0.915$

Step 2: Calculate Expected Frequencies using the Poisson Formula

Using the calculated mean ($\lambda = 0.915$), we can now use the Poisson formula to find the theoretical probability ($P(X = x)$) for each number of failures. The expected frequency for each $x$ is then obtained by multiplying this probability by the total number of observations (200 machines).

The formula to use is: $P(X = x) = \frac{0.915^x e^{-0.915}}{x!}$

Step 3: Tabulate Observed and Expected Frequencies

We now create a table comparing the observed frequencies with the expected frequencies.

| No. of System Failures ($x$) | Observed Frequency ($f$) | $fx$ | $P(X = x)$ (Calculated) | Expected Frequency ($P \times 200$) | | :------------------------- | :----------------------- | :---- | :---------------------- | :---------------------------------- | | 0 | 80 | 0 | $\approx 0.4005$ | $\approx 80.10$ | | 1 | 72 | 72 | $\approx 0.3667$ | $\approx 73.34$ | | 2 | 38 | 76 | $\approx 0.1677$ | $\approx 33.54$ | | 3 | 6 | 18 | $\approx 0.0512$ | $\approx 10.24$ | | 4 | 3 | 12 | $\approx 0.0117$ | $\approx 2.34$ | | 5 | 1 | 5 | $\approx 0.0021$ | $\approx 0.42$ | | Total | 200 | 183 | | $\approx 200.00$ |

(Note: Individual expected frequencies are rounded for clarity.)

Conclusion

The expected frequencies calculated using the Poisson distribution with $\lambda = 0.915$ are very close to the actual observed frequencies. This strong agreement indicates that the Poisson distribution is an appropriate model for the number of system failures in this factory setting. The data suggests that the number of system failures can be reliably modeled by a Poisson process with an average rate of 0.915 failures per machine per week.

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Interview Questions

• What does it mean to fit a Poisson distribution to data?

• How do you calculate the mean ($\lambda$) when fitting a Poisson distribution?

• What is the role of the Poisson formula in the model fitting process?

• How are expected frequencies derived using Poisson probabilities?

• What does a close match between observed and expected frequencies imply?

• Why is the Poisson distribution particularly suitable for modeling rare events?

• In what real-world scenarios is the Poisson distribution commonly applied?

• How would you interpret a $\lambda$ value of less than 1 in a Poisson model?

• What are the key assumptions that must hold true to effectively use the Poisson distribution?

• Can you explain the step-by-step process to validate a Poisson fit for a dataset?