The generalised Pareto distribution is useful in modelling extreme value problems or rare events....1 answer below »

1. The generalised Pareto distribution is useful in modelling extreme value problems or rare events. For example, it can be used to model geophysical phenomena such as floods or extreme windstorms. A random variable X is said to have a generalised Pareto distribution with parameter > 0 if its CDF is Fx(x) = 1 — (1+ x)-11°, x > 0. (a) Let Y = o ln(1 + X). Obtain the PDF of Y. Identify the distribution of Y. (b) Let W = • Obtain the PDF of W. (Reduce to its simplest form.) 2. The marginal distributions of X and Y being normal doesn't imply that the joint distri-bution of (X, Y) is bivariate normal. Let (X, Y) has joint PDF f(x,v)= exP 2 — (x2 + y2) ( 2 1 + xy exp — 2 (x2 + 1/2 - 2) }) 7r where —oo (a) Show that the marginal PDFs of X and Y are normal. (b) Find the covariance and correlation coefficient of X and Y. 3. The geometric mean of a sequence of positive numbers xi, ... x„ is c/xi x„. Let X1 i ... X„ be a random sample on an exponential distribution with mean 0. We would like to use the geometric mean G = X., to estimate 0. Obtain an unbiased estimator of 0 based on G. Your answer may be expressed in terms of the gamma function CO 1'(z) = 4. Sometimes calculus is not helpful in obtaining maximum likelihood estimators. Let Xi, X2, .. , X. be a random sample from a uniform distribution on [0, 0]. Suppose results xi, x2, . , x,, are observed. Since 1(x). 1/8 for 0 1 L(0) _= 0+„ 0 xi 0,...,0 x,, 0 0, otherwise. As long as 8 > max(x,), the likelihood is 1/0", which is positive, but as soon as 8

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Hi, Kindly go through the attachment. It a scanned file in pdf which includes the complete answer of the question asked. A detailed approach is taken while writing the report, If in case there is any doubt or ambiguity kindly mark it and revert it back. Moreover, there is always a possibility of error, so I am always here to help with edits and revisions. Based on the provided solution, do rate it and give a positive feedback. Genuine criticism is encouraged as this is how we learn and progress. Thanks