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Project 16: Poisson Distribution Simulation

Problem Statement

Simulate n values of a Poisson random variable X = poisson(λ) (choose λ > 0). Compute the sample mean (x̄), sample median (m), and sample standard deviation (s). Plot these quantities as functions of n (following the "General remark" for incremental generation). Observe the limit values these statistics converge to as n → ∞. What are these limits? Compare with theory. Estimate the variance of x̄ and m for n = 100 using 10,000 repetitions to determine which estimate is better (smaller variance).

Approach

  1. Parameters:

    • λ (lambda, rate parameter of Poisson): Choose a value, e.g., λ = 10.
    • max_n: Maximum sample size for convergence plots, e.g., 10,000.
    • step: Increment for n in convergence plots, e.g., 100.
    • n_variance: Sample size for variance estimation, e.g., n = 100.
    • num_repetitions: Number of repetitions for variance estimation, e.g., 10,000.

  2. Simulation (Incremental):

    • Similar to Project 1, generate Poisson samples incrementally.
    • Start with n = step samples from poisson(λ).
    • For subsequent n values (2*step, 3*step, ..., max_n), add step new samples to the existing dataset.
    • Use numpy.random.Generator.poisson(lam=λ, size=...).

  3. Statistics Calculation:

    • For each value of n in the incremental simulation, calculate:
      • Sample Mean: numpy.mean()
      • Sample Median: numpy.median() (Note: Poisson is discrete, median will be an integer or half-integer).
      • Sample Standard Deviation: numpy.std(ddof=1).

  4. Plotting Convergence & Theoretical Limits:

    • Plot Sample Mean vs. n, Sample Median vs. n, Sample Standard Deviation vs. n on separate plots.
    • Theory: For X ~ poisson(λ):
      • E[X] (True Mean) = λ
      • Var(X) (True Variance) = λ
      • True Standard Deviation = sqrt(λ)
      • The median of a Poisson distribution is approximately floor(λ + 1/3 - 0.02/λ) for large λ, typically close to λ.
    • Add horizontal lines on the plots representing the theoretical limits (λ for mean/median, sqrt(λ) for std dev) to check convergence.

  5. Convergence Comparison (Variance Estimation):

    • Similar to Project 1, set n = n_variance (e.g., 100).
    • Repeat num_repetitions (10,000) times:
      • Generate a new sample of n values from poisson(λ).
      • Calculate the sample mean (x̄) and sample median (m).
    • Collect the 10,000 means and 10,000 medians.
    • Compute the sample variance of the collected means and medians.
    • Compare the variances. The estimator with the smaller variance is better (more efficient) for this n.
    • Theoretical Variance: Var(x̄) = Var(X)/n = λ/n. There isn't a simple formula for Var(m) for Poisson, but we expect it to be larger than Var(x̄).

  6. Implementation: project_16.py and test_project_16.py. Include visualizations for the convergence plots.