Use the inverse transform method to generate a sample of the exponential distribution with parameter θ, which is given by its density
1/θe-x/θ for x ≥ 0
f(x) =
0 for x < 0
where θ is a positive parameter, that is θ > 0.
Provide the following in BTEX-format in a pdf-file:
1. The cumulative distribution function F (including derivation).
2. Its inverse, F-1 (with proof).
3. For the particular choices θ = 2 and 10, create histograms for sample sizes n = 100, 1,000, and 10,000. Explain what you expect to see and what you actually see. Give a justification if there is a difference between your expectation and what you actually see.
4. For an exponential distributed random variable X, calculate the theoretical mean,
μ = E[X],
and variance
σ2 = E[(x - μ)2] = E[x2] - E[X]2.
5. Use a sample of the exponential distribution to estimate μ and σ2. Using these estimations to write down the 66.66%, the 90%, and the 99% asymptotical confidence interval for the estimate of μ. Do all of this for the sample sizes n = 100, 1, 000, and 10, 000. Explain what you expect to see and what you actually see. Give a justification if there is a difference between your expectation and what you actually see.