Let X₁, X₂, X3,..., Xn be a random Sample from N (μ,0²) Use maximum likelihood estimation to find mean and variance 0² (SHOW DERIVATION STEPO) Verify the maximum likelihood estimator is unbiased Use the method of moment to find the mean and variance.
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- Let X1, X2,…. Xn be a random sample from Poisson distribution with parameter , obtain Cramer Rao lower bound for the variance of unbiased estimators of lamdaSuppose that X1, . . . , Xn is a random sample from the Normal distribution N (0, σ2 ) with parameter σ > 0, Find the Maximum Likelihood Estimation of σ.Let X1, ..., Xn be a sample from a geometric random variable with parameter p.(a) Find the maximum likelihood estimator for p.(b) Is the estimator unbiased?(c) Is the estimator consistent?
- Let us consider a discrete random variable having the pmf given by, PX(k) ={ (1/3) , k = 1 (2/3) , k = 2. 0 , k = 3 Calculate the moment generating function for X. Also get the values of its mean and variance.Let X1, X2, …, Xn be a random sample from the Normal distribution N() (a) Using method of moments to estimate the parameters and . (b) Are those estimators unbiased?Let X1, ..., Xn be a sample from an Poisson population with parameter λ.(a) Find the maximum likelihood estimator for λ.(b) Is the estimator unbiased?(c) Is the estimator consistent?