Question

Problem 1 - Assume that X is exponentially distributed with rate λ = 0.2. We are interested in
computing E[(X - 3)⁺]. (Note: a⁺ = Max(a, 0), i.e, if a < 0 then a⁺ = 0 and if a>=0 then a⁺ = a.)

(a) Compute the expected value exactly.
(b) Estimate the expected value using Monte Carlo simulation and provide a 95% confidence interval
for your estimate. Note: You can generate a random sample of an exponentially distributed random
variable with rate in Numpy using np.random.exponential(1/λ).
(c) Create a plot that demonstrates the convergence of the Monte Carlo estimate to the exact value
as the number of samples increases.

Answer 1