Maximum Likelihood Estimation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Making the data most likely

Mathematics · Data & Statistics · ages 20-21
Name ______________________   Date ____________

  1. Three trials come out success, success, failure in that order. Which likelihood function of p fits?

    • p squared times (1 minus p)
    • p cubed
    • (1 minus p) cubed

  2. The likelihood is the chance of the observed data, written as a function of the parameter.

    Circle one:   True   False

  3. After differentiating the log-likelihood, which pair of checks confirms a maximum?

    • Positive slope plus zero curvature
    • Zero slope plus negative curvature
    • Zero value plus positive curvature

  4. Three Bernoulli trials come out success, success, failure in that order. Which likelihood function of p fits?

    • p squared times (1 - p)
    • p cubed
    • (1 - p) cubed
    • p times (1 - p)

  5. Lifetimes look exponential with sample mean 4 years. Estimate the rate lambda by maximum likelihood.

    Answer: ______________

  6. Thirty trials yield 12 successes. Compute the maximum likelihood estimate of the success chance p.

    Answer: ______________

  7. The log-likelihood has zero slope and negative curvature at p equal to 0.4. Jo claims p equal to 0.4 is the maximiser. Is Jo right?

    Circle one:   True   False

  8. The log-likelihood has zero slope and negative curvature at p = 0.4. Jo claims p = 0.4 is the maximiser. Is Jo right?

    Circle one:   True   False

  9. A log-likelihood for a rate has slope 8 minus 32 lambda. Setting the slope to zero, what is the maximiser?

    Answer: ______________

  10. A student maximises the likelihood directly on a big sample, matches the log route answer, and concludes logs were pointless. What is wrong?

    • Logs move the peak, so the answers should differ
    • Nothing, logs never matter
    • Logs turn products into sums, which is what made the differentiation routine
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Making the data most likely W1-mt_ZRTrcy7-KV-s1

  1. p squared times (1 minus p) · Ordered outcomes multiply: p for each success and (1 minus p) for the failure.
  2. True · That reading of the joint chance as a function of the parameter is the definition.
  3. Zero slope plus negative curvature · Zero slope marks a stationary point and negative curvature marks the peak.
  4. p squared times (1 - p) · Ordered outcomes multiply: p for each success and (1 - p) for the failure, giving p x p x (1 - p).
  5. 0.25 · The exponential MLE is one over the sample mean: 1/4 = 0.25.
  6. 0.4 · The binomial MLE is the sample share: 12 / 30 = 0.4.
  7. True · Zero slope plus downward bending confirms the peak, so Jo is right.
  8. True · True. Zero slope marks a stationary point and negative curvature marks a peak, together confirming a maximum.
  9. 0.25 · Zero slope gives 8 equal to 32 lambda, so lambda is 8 over 32.
  10. Logs turn products into sums, which is what made the differentiation routine · Logs simplify the calculus while leaving the maximiser unchanged.
Worksheet · LightMySky