Let X1, X2, . . . , Xn be a random sample of size n from...

Let X1, X2, . . . , Xn be a random sample of size n from a Poisson distribution with unknown mean µ. It is desired to test the following hypotheses

H0 : µ = µ0 versus H1 : µ not equal to µ0

where µ0 > 0 is a given constant. Derive the likelihood ratio test statistic

Expert Solution

orchestra answered 1 year ago

Let X1, X2, ..., Xn be a random sample of size from a distribution with probability...

Let X1, X2, ..., Xn be a random sample of size from a distribution with probability density function f(x) = λxλ−1 , 0 < x < 1, λ > 0 a) Get the method of moments estimator of λ. Calculate the estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3. b) Get the maximum likelihood estimator of λ. Calculate the estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.

. Let X1, X2, ..., Xn be a random sample of size 75 from a distribution...

. Let X1, X2, ..., Xn be a random sample of size 75 from a distribution whose probability distribution function is given by f(x; θ) = ( 1 for 0 < x < 1, 0 otherwise. Use the central limit theorem to approximate P(0.45 < X < 0.55)

Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval...

Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval [θ1,θ2]. Let Y = min (X1,X2,...,Xn). (a) Find the density function for Y. (Hint: find the cdf and then differentiate.) (b) Compute the expectation of Y. (c) Suppose θ1= 0. Use part (b) to give an unbiased estimator for θ2.

Let X1, ..., Xn be a random sample of size n from an exponential ditribution with...

Let X1, ..., Xn be a random sample of size n from an exponential ditribution with parameter lambda . Let Di=(n-i+1)[X(i)-X(i-1)] i=1,.... n , where X(0)=0 be the normalized spacings. Using Jacobians, derive the joint distribution of D1,....., Dn.

: Let X1, X2, . . . , Xn be a random sample from the normal...

: Let X1, X2, . . . , Xn be a random sample from the normal distribution N(µ, 25). To test the hypothesis H0 : µ = 40 against H1 : µne40, let us define the three critical regions: C1 = {x¯ : ¯x ≥ c1}, C2 = {x¯ : ¯x ≤ c2}, and C3 = {x¯ : |x¯ − 40| ≥ c3}. (a) If n = 12, find the values of c1, c2, c3 such that the size of...

. Let X1, X2, . . . , Xn be a random sample from a normal...

. Let X1, X2, . . . , Xn be a random sample from a normal population with mean zero but unknown variance σ 2 . (a) Find a minimum-variance unbiased estimator (MVUE) of σ 2 . Explain why this is a MVUE. (b) Find the distribution and the variance of the MVUE of σ 2 and prove the consistency of this estimator. (c) Give a formula of a 100(1 − α)% confidence interval for σ 2 constructed using the...

Let X1,X2, . . . , Xn be a random sample from the uniform distribution with...

Let X1,X2, . . . , Xn be a random sample from the uniform distribution with pdf f(x; θ1, θ2) = 1/(2θ2), θ1 − θ2 < x < θ1 + θ2, where −∞ < θ1 < ∞ and θ2 > 0, and the pdf is equal to zero elsewhere. (a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint sufficient statistics for θ1 and θ2, are complete. (b) Find the MVUEs of θ1 and θ2.

Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution...

Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; θ) = (x/θ) * e^x^2/(2θ) for x > 0 (a) It can be shown that E(X2 P ) = 2θ. Use this fact to construct an unbiased estimator of θ based on n i=1 X2 i . (b) Estimate θ from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions: 16.88 10.23 4.59 6.66...

Let X1, X2, … , Xn be a random sample from a normal population with zero...

Let X1, X2, … , Xn be a random sample from a normal population with zero mean and unknown variance σ^2 . Find the maximum likelihood estimator of σ^2

Let X1, X2, ..., Xn be a random sample from Exp(?). Find the MVUE of the...

Let X1, X2, ..., Xn be a random sample from Exp(?). Find the MVUE of the median of this exponential distribution.

Question