Ms455 ms555project 1 - simulation in finance for the

Question

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.