Losses are Normal with u 300 and o = 10. Determine the 90% Tail Value at Risk. Hint: For the Normal Distribution, E[X A x] = u D((x-u)/o]- o o((x-)/0] + x {1 -Þ[(x-}.
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- A Troublesome Snowball One winter afternoon, unbeknownst to his mom, a child bring a snowball into the house, lays it on the floor, and then goes to watch T.V. Let W=W(t) be the volume of dirty water that has soaked into the carpet t minutes after the snowball was deposited on the floor. Explain in practical terms what the limiting value of W represents, and tell what has happened physically when this limiting value is reached.Consider the joint PDF f(x,y)=4xy for 0<x<1 and 0<y<1. (a) Are X and Y independent?(b) Find both marginal distributions. (c) Find both conditional distributions.The moment generating function can be usedto find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X)=meanand V(X) =varianceb. Repeat (a) using RX(t)=ln[MX(t)], and comparewith part (a) in terms of effort.
- Let X|Λ have an exponential distribution with conditional survival function SX|Λ(x|λ) = e−λx. Let Λ have a Gamma distribution with parameters α and θ. Find the unconditional or marginal survival function of X.(a) Determine the value of k for a valid pdf.(b) Complete the cdf of X .(c) What is the expected total deduction on a randomly chosen form?(d) Let Y = ln(X /5), where ln(·) is natural logarithm. Determine the pdf as wellas the cdf of Y , and further identify which common distribution that Y follows?solveIn random sampling from the exponential distribution, f(x) =1 θθe x− , x > 0, θ> 0, find the maximum likelihood estimator of θ and obtain the asymptotic distribution of this estimator
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