Fitting a Binomial Distribution 🧩

Fitting means calculating the Theoretical (Expected) Frequencies for a given observed frequency distribution. We assume the data follows a Binomial law and calculate what the frequencies should be.

Steps to Fit 🚶

1. Find Mean: Calculate the actual arithmetic mean (X̄) of the given data. Mean = ∑ fX / ∑ f (for frequency data).
2. Estimate Parameters:
   • Set Mean = n * p.
   • Find p = Mean / n.
   • Find q = 1 - p.
3. Calculate Probabilities: Use P(r) = nCr p^r q^(n-r) for r = 0, 1, 2... n.
4. Expected Frequencies: Multiply each probability by total frequency N. Expected Freq = N * P(r)

Example 📝

Given: 4 coins are tossed 80 times. Number of heads (X) and Frequency (f) are given. X: 0, 1, 2, 3, 4 f: 4, 18, 30, 20, 8 (N=80)

Fit a Binomial Distribution.

Step 1: Find Mean Mean = (0*4 + 1*18 + 2*30 + 3*20 + 4*8) / 80 Mean = (0 + 18 + 60 + 60 + 32) / 80 Mean = 170 / 80 = 2.125

Step 2: Find p and q n = 4 (4 coins). Mean = np => 2.125 = 4p p = 2.125 / 4 = 0.53 q = 1 - 0.53 = 0.47

Step 3 & 4: Calculate Expected Frequencies Using N * nCr * p^r * q^(n-r).

• r=0: 80 * 4C0 * (0.53)^0 * (0.47)^4
• r=1: 80 * 4C1 * (0.53)^1 * (0.47)^3
• ...and so on.

Note: For perfect unbiased coins, we assume p = 0.5 directly. Usually questions specify if coins are unbiased.

Recurrence Formula (Shortcut) ⚡

To avoid calculating nCr every time:

P(r+1) = [ (n-r) / (r+1) ] * (p/q) * P(r)

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