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Test whether samples follow a binomial distribution

J.P. MorganData Scientist · Onsite
MediumData ScientistOnsiteMachine Learning

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This problem tests the application of goodness-of-fit testing to verify if a set of discrete observations conforms to a binomial distribution.

Problem Statement

You are working on a risk modeling team and receive a dataset containing the number of successful outcomes across N independent trials for M different experiment groups. Each experiment group consists of n trials with a success probability p.

Your task is to determine whether the observed success counts in these M groups follow a binomial distribution B(n, p).

Requirements

  1. Given the array observations of length M (where each element is the number of successes in a group of n trials), estimate the parameter p using the Method of Moments.
  2. Construct the expected frequency for each possible outcome k (where 0 <= k <= n) under the null hypothesis that the data follows B(n, p).
  3. Propose and implement a statistical test to quantify the goodness-of-fit. Provide the test statistic and explain how to determine the p-value.

Example

  • Input: observations = [1, 2, 2, 3, 1], n = 4, M = 5
  • Step 1: Mean successes = (1+2+2+3+1)/5 = 1.8. Estimate p = 1.8 / 4 = 0.45.
  • Step 2: Calculate expected frequencies for k = 0, 1, 2, 3, 4 using P(X=k) = C(n, k) * p^k * (1-p)^(n-k) multiplied by M.
  • Step 3: Compare expected vs. actual frequencies using the Chi-Squared test.

Constraints

  • n (trials per group) is between 5 and 50.
  • M (number of groups) is at least 100.
  • Ensure all expected bin frequencies are greater than 5 to maintain the validity of the Chi-Squared approximation.

Hints

Hint 1

Use the sample mean to estimate the probability p, knowing that for a binomial distribution, the mean is E[X] = n * p.

Hint 2

The Chi-Squared goodness-of-fit test is appropriate here, where the test statistic is the sum of (Observed - Expected)^2 / Expected.

Hint 3

Remember to subtract degrees of freedom for the parameter estimated from the data when calculating the p-value from the Chi-Squared distribution.

Follow-up questions

  • How would your approach change if you were testing for a Poisson distribution instead of a Binomial distribution?
  • What are the limitations of the Chi-Squared test if your expected frequencies in certain bins are very small?
  • If the test rejects the null hypothesis, how would you investigate whether the data is 'overdispersed' (i.e., variance is higher than the binomial model predicts)?