STAT213 2019 exam potential solutions (1)

Q.1 [3 marks]: Macro Question
Q.2 [4 marks]: Macro Question
- Probability Mass Function (PMF):
  - Given: f(x) = 3x²/k for x = 0, 1, 2, 3
  - Find k
  - Cumulative Distribution Function (CDF):
  - F(x) = {1 - (27/x*) for x < 3; F(x) = 1 for x ≥ 3}
  - Tasks:
    - (a) Find k.
    - (b) Find F(x).
    - (c) Compute P(X < 6).
    - (d) Find E(X).
    - (e) Compute P(X > 1).
    - (f) Determine E(2X² - 2X - 3).
    - (g) Find Var(2X - 42) = 3(1 + 4 + 9).

Q.5 [4 marks]: Another Macro Question
Q.6 [5 marks]: Macro Question (Note: Multinomial distribution not covered)
- Probability Density Function (PDF) of X:
  - Proportions of genotypes: AA = θ², Aa = 2θ(1-θ), aa = (1-θ)² for 0 < θ < 1.
  - Given sample: {0.3, 0.5, 0.1, 0.7}.
  - Tasks:
    - Estimate θ using Method of Moments.
    - Write the likelihood function.
    - Obtain log-likelihood function, score function, and observed information.
    - Given n1 = 128, n2 = 24, n3 = 8, find the maximum likelihood estimate for θ

Q.7 [5 marks]: Macro Question
Q.8 [6 marks]:
- Given a random sample from a gamma population:
  - PDF for success probability θ: f(x;θ) = θ(1-θ)^(1-x) for x=0,1.
  - Tasks to prove:
    - Show that MLE for θ works.
    - Likelihood: Use logarithmic forms
    - Derivatives with respect to likelihood to find MLE.
    - Check 2nd derivative for maxima.
Note: No Bayesian question or statistical testing question present in the examination.