Let X1, X2, . . . , Xn be iid Poisson random variables with unknown mean...

Let X₁, X₂, . . . , Xn be iid Poisson random variables with unknown mean µ

1. Find the maximum likelihood estimator of µ

2.Determine whether the maximum likelihood estimator is unbiased for µ

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a)...

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a) Find the MVUE for λ. (b) Find the MVUE for λ 2

Let X1, X2, . . . be iid random variables following a uniform distribution on the...

Let X1, X2, . . . be iid random variables following a uniform distribution on the interval [0, θ]. Show that max(X1, . . . , Xn) → θ in probability as n → ∞

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter...

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter µ. a. What is the mean and variance of this distribution? b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator? Why or why not? c. Find the moment estimator and MLE of µ. d. Show the estimators in (c) are unbiased. e. Find the MSE of the estimators in (c). Given the frequency table below: X 0...

Let X1, X2, . . . , Xn be iid following a uniform distribution over the...

Let X1, X2, . . . , Xn be iid following a uniform distribution over the interval (θ, 2θ) (θ > 0). (a) Find a method of moments estimator of θ. (b) Find the MLE of θ. (c) Find a constant k such that E(k ˆθ) = θ. (d) By using the Rao-Blackwell, which estimators of (a) and (b) can be improved?

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on...

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on the interval [0, θ]. (a) Find the pdf of X(j) , the j th order statistic. (b) Use the result from (a) to find E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the mean difference between two successive order statistics. (d) Suppose that n = 10, and X1, . . . , X10 represents the waiting times that the n = 10...

Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n,...

Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is independent of X1, X2, . . .. Further define T = XN . (a) Find E(T) and argue that T is short proper. (b) Find the pgf of T. (c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0. (d) Use the pgf of...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine a) the MLE ? of ?. b) E(? ̂). c) the asymptotic variance of the MLE of ?. d) the MLE of SD(Xi ) = √ ?.

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2)...

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ. Determine a) L(?1, ?2). b) the MLE of ?⃗ = (?1, ?2). c) E(? ̂ 2).

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses...

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses H0 : µ = µ0 and H1 : µ (not equal)= µ0 and the test statistic (X bar − µ0)/ (S/√ n). Note that S has been used as σ is unknown. a. What is the distribution of the test statistic when H0 is true? b. What is the type I error of an α−level test of this type? Prove it. c. What is...

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) =...

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b < 1 then, find a two dimensional sufficient statistic for (a, b)

Subjects