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- write the log likelihood for the null model and the number of parameters associated with the maximal modellet x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x isLet X and Y be independent exponential random variables with parameter 1. Find the cumulativedistribution function of Z = X/(X + Y )
- Suppose that the random change in value of a financial asset is X over the first day and Y over the second. Suppose also that Var(X) =18 and Var(Y) = 26 In this case, the total change in the value over these two days is given by X +Y. Do you have enough information to compute Var(X +Y)? If so, compute this value. If not, explain what additional information you need to do so.let k be a random variable that takes with equal prob 1/(2n+1), values in integer the interral' [-n, n]. Find the PMF of the random variable y = log (x) where X = a¹" and a>o.2. Let Y,,., Y, be independent random variables such that Y, (Yı., Yp)" and 0 = (0,.,0p)". Let = 0(Y) = (0,(Y),... , @p(Y))" be an estimator of 0, and let g(Y) = (g(Y),... , gp(Y))" = – Y. Denote by || - || the Euclidean norm, ||Y° = Y} + .. + Y. N(8), 1). Write Y = %3D Suppose that D(Y) = @g(Y)/ay, exists. Then it is known that %3D R(Ô(Y)} = +2 D(Y) + É19(Y)² =1 is an unbiased estimator of the risk of 0, under squared error loss L(0, ê) = ||0 – e|P. [You are not required to show this]. %3D (i) The James-Stein estimator is 6.s(Y) = (1 – )Y. _P-2y ||Y? Show that an unbiased estimator of the risk of d Js(Y) is Řlójs(Y)) = p – (p - 2) /||YII°. Deduce that Y is inadmissible as an estimator of 0. Is ô js(Y) admissible? Justify your answer.
- Suppose the time it takes Alex to do this exam is exponentially distributed with parameter 3 per hour, and the time it takes Ben to do the exam is exponentially distributed with parameter 2per hour. Assume that these two times are independent.(a) What is the probability that Alex finishes before Ben?(b) What is the expected time in minutes until the first one finishes this exam?(c) What is the probability that neither Alex nor Ben finishes the exam within 3 hours?Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).Let X and Y are 2 independent random variables N (0,1) Questions: how to find E(X), E(Y), E(X^2), E(Y^2)? Thank you
- Let X be a random variable with the function, -(x -A) "for x > 2, Let f (x) = { elsewhere. Derive u and show that T = X is biased estimator of 2 . Can you modify T =X in order to get an unbiased estimator of 2.Prove that if M(t) is the MGF of a random variable X, then the MGF of a + bX is e^at M (bt)Suppose that X₁, the number of particles emitted in t hours from a radio - active source has a Poisson distribution with parameter 20t. What is the probability that exactly 5 particles are emitted during a 15 minute period?