Let (x1, x2) be i.i.d. samples from a Poisson distribution with unknown parameter > 0. Suppose that we wish to test Ho A = 3 versus H₁ : X < 3. a (a) Consider the test which rejects for T = (x₁+x₂) < 2. What is the level of significance a? (b) Find the power of this test for λ = 2.
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- Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?Find the sampling distributions of Y1 and Yn for ran-dom samples of size n from a population having the beta distribution with α = 3 and β = 2.A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?
- Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxyLet X1, X2, ..., X10 be a random sample from a gamma distribution with α = 3 and β = 1/θ. Suppose we believe that theta follows a gamma-distribution with α = 28 and β = 17 and suppose we have a trial (X1 , ... , Xn ) with an observed x̄ = 28.2 What is the posterior distribution of θ What is the Bayes point estimate of θ associated with the square-error loss function?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 exponential population with parameter λ.(a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased?(c) Is the estimator consistent?Suppose X1, . . . , Xn be a random sample from the Beta(θ, 1) distribution. Find the P-value for the LRT test of the hypotheses H0 : θ ≥ 1 vs H1 : θ < 1Let 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?
- If X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal placesIf the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.Suppose 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.