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Assess a Distribution and Derive Coin-Toss Expectations

CargurusData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This problem tests the ability to distinguish between Bernoulli and Binomial processes, calculate expected values and variances, and validate model assumptions based on observed sample data.

Part 1: Coin Toss Modeling

You are given a sequence of 100 independent coin flips, where 65 resulted in 'Heads'.

  1. Calculate the Maximum Likelihood Estimate (MLE) for the probability of landing Heads, denoted as p_hat.
  2. If this coin were tossed 10 times, what is the expected number of Heads? Provide the variance of this distribution.
  3. Identify the specific distribution being used and state the underlying assumptions required for this model to be valid.

Part 2: Distribution Diagnostics

You receive a new dataset containing 1,000 observations of the number of 'Heads' occurring in batches of 10 tosses each. Upon inspection, the sample mean is 5.0, but the sample variance is 9.0.

  1. Does this data follow a standard Binomial distribution? Why or why not?
  2. If the Binomial model fails to describe the variance, what phenomenon does this suggest, and how would you adjust your modeling approach to account for it?

Constraints & Considerations

  • Assume the coin tosses in Part 1 are independent and identically distributed (i.i.d).
  • For Part 2, justify your reasoning using the properties of the Binomial distribution Var(X) = n * p * (1 - p).
  • Consider edge cases: What happens if the sample size in Part 1 was only 2 flips?

Hints

Hint 1

Recall that for a Binomial distribution with parameters n and p, the mean is n*p and the variance is n*p*(1-p).

Hint 2

Compare the calculated variance of a Binomial distribution (given the mean) against the observed variance of 9.0 in Part 2.

Hint 3

When variance exceeds the mean in count data, consider looking into overdispersion models like the Beta-Binomial distribution.

Follow-up questions

  • How would you conduct a formal hypothesis test to determine if the coin is fair?
  • If the coin flips were not independent (e.g., a 'streak' effect), how would that impact your variance calculation?
  • How does the choice of a prior distribution in a Bayesian framework influence the estimation of p when the sample size is very small?