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If X follows a two-parameter Pareto distribution with a = 3 and θ = 100, find...
12. Suppose XIX, iid X, P(θ, l), where P(0,1) is the one-parameter Pareto distribution with density f(x)-0/10+1 for l < x < 00, Assume that θ >2, so that the model θ/(0-1)(8-2)2 (a) obtain the MME θι from the first moment equation and the MIE θ2 (b) Obtain the asymptotic distributions of these two estimators. (c) Show that the ML is asymptotically superior to the MME P(0,1) has finite mean θ/(9 -1 ) and variance
The random variable X is distributed as a Pareto distribution with parameters α = 3, θ. E[X] = 1. The random variable Y = 2X. Calculate V ar(Y )
I. Let X be a random sample from an exponential distribution with unknown rate parameter θ and p.d.f (a) Find the probability of X> 2. (b) Find the moment generating function of X, its mean and variance. (c) Show that if X1 and X2 are two independent random variables with exponential distribution with rate parameter θ, then Y = X1 + 2 is a random variable with a gamma distribution and determine its parameters (you can use the moment generating...
2. Suppose that X|θ ~ U(0.0), the uniform distribution on the interval (09). Assuming squared error loss, derive that Bayes estimator of θ with respect to the prior distribution P(α.θο), the two-parameter Pareto model specified in (3.36), first by explicitly deriving the marginal probability mass function of X, obtaining an expression for the posterior density of θ and evaluating E(θ x) and secondly by identifying g(θ|x) by inspection and noting that it is a familiar distribution with a known mean.
let X=pareto(α,γ) find the distribution and density function of Y=logX
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
5. For X follows Exp(6) (exponential distribution with parameter θ), a hypothesis test rejects the null hypothesis Ho : θ-1 when X k versus H1 : θ > 1. (a) Show that for any k greater than -log(0.05), the test has the probability of type I error less than 0.05 (b) Show that the power of the test at θ-10 is larger when k-1 than k-2. (c) Let k-_ log(0.05), calculate the power function in terms of θ when θ...
The random variable X follows a two-parameter Pareto with coefficient of variation 1.5. Determine the coefficient of skewness of X .
3. X is the random variable for claim sizes. Given A, X follow a single-parameter Pareto distribution with parameters θ 1000 and A. The distribution of A over the entire population is an exponential distribution with mean 3 Calculate Pr(X> 1500)
3. If X follows an exponential distribution with mean 1/λ. Find the density function of Y, where (b) Y = 1/x.