Fitting Binomial Distribution
Introduction
- Fitting a binomial distribution involves comparing observed frequencies with expected frequencies derived from the binomial probability formula.
- The recurrence relation simplifies the process of finding probabilities.
- This technique is useful for testing if a dataset follows a binomial distribution.
Binomial Probability Function
The binomial probability function is:
$$p(x) = {n \choose x} p^x q^{n – x}$$
Where:
- n = number of trials
- p = probability of success
- q = 1 – p = probability of failure
Derivation of Recurrence Relation
Start with the binomial probability function:
$$p(x) = \frac{n!}{x! (n – x)!} p^x q^{n – x}$$
For \( p(x + 1) \):
$$p(x + 1) = \frac{n!}{(x + 1)! (n – x – 1)!} p^{x+1} q^{n – x – 1}$$
Divide \( p(x + 1) \) by \( p(x) \):
$$\frac{p(x + 1)}{p(x)} = \frac{(n – x)}{(x + 1)} \cdot \frac{p}{q}$$
Therefore, the recurrence relation is:
$$p(x + 1) = \frac{(n – x)}{(x + 1)} \cdot \frac{p}{q} \cdot p(x)$$
Example: Tossing 4 Coins
Observed data:
- 0 heads: 15 times
- 1 head: 35 times
- 2 heads: 90 times
- 3 heads: 40 times
- 4 heads: 20 times
Total number of tosses: 200
Calculation of Mean and Probability
Mean:
$$\bar{x} = \frac{415}{200} = 2.075$$
Probability of success:
$$p = \frac{2.075}{4} = 0.5188, \quad q = 1 – p = 0.4812$$
Calculate:
$$p(0) = q^4 = (0.4812)^4 = 0.0536$$
Using Recurrence Relation
Use:
$$p(x + 1) = \frac{(n – x)}{(x + 1)} \cdot \frac{p}{q} \cdot p(x)$$
Calculate:
- p(1) = 4 × (0.5188 / 0.4812) × 0.0536 = 0.23115
- p(2) = (3 / 2) × (0.5188 / 0.4812) × 0.23115 = 0.37382
- p(3) = 2 × (0.5188 / 0.4812) × 0.37382 = 0.26859
- p(4) = (1 / 4) × (0.5188 / 0.4812) × 0.26859 = 0.0724
Detailed Summary Table
| Number of Heads (X) | Recurrence Relation | p(x) | Expected Frequency |
|---|---|---|---|
| 0 | Initial: q^4 | 0.0536 | 10.72 |
| 1 | 4 × (p/q) × p(0) | 0.23115 | 46.23 |
| 2 | (3/2) × (p/q) × p(1) | 0.37382 | 74.76 |
| 3 | 2 × (p/q) × p(2) | 0.26859 | 53.71 |
| 4 | (1/4) × (p/q) × p(3) | 0.0724 | 14.48 |
Conclusion
-
- The expected frequencies using the binomial model are close to the observed frequencies.
- The example supports the assumption that the coin is unbiased.
- The recurrence relation simplifies stepwise calculation of probabilities.
Video
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