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- 2.3 Explain how to generate from the Weib(a, A) distribution using the inverse- transform method.Q4. Let X1,X 2,.X , be a random sample from an exponential distribution with parameter 0. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of e. (b) Find the maximum likelihood estimator (m.l.e.) of 0. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? (d) What is the method of moments estimator of 0?Q11. Let X1,X 2,.,X, be a random sample from a Poisson distribution with parameter 2. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of 1. (b) Find the maximum likelihood estimator (m.l.e.) of å. (c) Is this m.l.e. unbiased and does its variance attain the Cramer-Rao lower bound?
- Vo)) 4663%9:39 PM You Just now (2) The two basic SDEs. Solve the following SDEs and find the expectation and variance of X₁. (i) dX, X,dt - 2XdB₁, Xo = 2. (ii) dx = -2X du +4dBu, 1 ≤u≤ 4, X₁ = 3.EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.Prob. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =
- 5. Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix.5. Find the method of moments estimator of a based on a random sample from a gamma distribution with B=2. XGAM(a,2) www.Q4. Let X1,X 2.X„ be a random sample from a Poisson distribution with mean parameter à. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of a. (b) Find the maximum likelihood estimator (m.l.e.) of 2. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? Justify your answer.
- Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.If X is a Poisson variable such that P(X =2) = 9P(X= 4) + 90P(X = 6), find the mean and variance of X.[Joint PDFS will be covered in Week 7] Let the random variables X and Y be the portions of the time in a day that two alternative routes between Topkapi and Uskudar have congestion (X for Route 1 and Y for Route 2). The joint PDF is given by fxy(x.y) = 2x2 + y? where Osxys1. (b) Assume Z=X+Y and W= XY and traffic experts are interested in the expected values of Z and W. E(Z) = and E(W) =O (Simplify your answers. Do not convert fractions into decimals.) (c) Find the variances of X and Y as well as the covariance between them. v(X) =D VY) =D and Cov(X, Y) =D (Simplify your answers. Do not convert fractions into decimals.) (d) The variance of Z can be found as V(Z) =- (Simplify your answer. Do not convert fractions into decimals.)