Fitting a Poisson Distribution 🧩

Similar to Binomial, we calculate theoretical frequencies assuming the data follows Poisson Law.

Steps to Fit 🚶

1. Find Mean: Calculate arithmetic mean of observed data. Mean (m) = ∑ fX / ∑ f.
2. Use Formula: P(r) = (e^-m * m^r) / r!
3. Calculate Expected Frequencies: N * P(r).

Recurrence Formula (Shortcut) ⚡

To determine probabilities quickly:

P(r+1) = (m / r+1) * P(r)

Procedure:

1. Calculate P(0) = e^-m.
2. Calculate P(1) = (m/1) * P(0).
3. Calculate P(2) = (m/2) * P(1).
4. And so on.

Example 📝

X: 0, 1, 2, 3 f: 123, 59, 14, 4 (N=200)

Step 1: Mean m = (0*123 + 1*59 + 2*14 + 3*4) / 200 m = (0 + 59 + 28 + 12) / 200 m = 99 / 200 = 0.495

Step 2: Probabilities Use m = 0.495. P(0) = e^-0.495. P(1) = m * P(0).

Step 3: Frequencies Multiply each P(r) by 200.

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