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.
Problem 1 - Assume that X is exponentially distributed with rate λ = 0.2. We are...
a) Let T be an exponentially distributed random variable with parameter l= 1. Let U be a uniformly distributed random variable. Use inversion to show how to calculate samples {tı, t2, ..} from samples {U1, U2, ..} of U. b) Use any software of your choice to estimate by Monte Carlo simulation: E[sin(tanh, LT))
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
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Let X be an exponentially distributed random variable with parameter λ. Prove that P(X > s + tK > t) P(X > s) for any S,12 0
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2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 11. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt < X <Y). (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? (e) Compute PIZ> t where t20. (f) Compute the pd.f. of Z. Z = min(X,Y)
Problem 1 [20 points X is an exponentially distributed random variable with parameter A. a, b with b >a 0 are real numbers. Find PLX > E [a, b))