代做STAT 400, Spring 2024 Homework 09代做Python语言

Homework 09

Due Tuesday, March 26, 11:59pm

STAT 400, Spring 2024

Each Exercise or lettered part of an Exercise is worth 5 points. Homework assignments are worth 50 points.

Exercise 1

Let X be a random variable that follows the Poisson distribution with mean λ, where λ > 0.

(a) What is the moment-generating function for X, a Poisson random variable? Derive the mgf using the definition and show your work.

(b) Find E[X] and Var[X] using the mgf of X found in part (a).

Exercise 2

Let X1, X2, … , Xn be an iid random sample of size n from a Poisson distribution with mean λ, where λ > 0.

(a) Find the method of moments estimator of λ, .

(b) Find the maximum likelihood estimator of λ, .

(c) Which of the estimators is unbiased for λ? The method of moments estimator (), the MLE (), neither, or both?

Exercise 3

Let X1, X2,…, Xn be an iid random sample of size n from an Exponential(θ) distribution with probability density function

(a) Find the maximum likelihood estimator for θ, . Then using that result, calculate the estimate when x1 = 14, x2 = 25, and x3 = 18. (This will be a number.)

(b) The mean squared error (MSE) of an estimator is defined as MSE() = [Bias()]2 + Var().

Calculate the value of the bias of the maximum likelihood estimator for θ, . (That will be a number.) Then calculate the mean squared error of . (This will be in terms of θ.)

Exercise 4

Let X1, X2,…, Xn be an iid random sample where Xi ~ Normal(µ, σ2), µ unknown, and σ2 unknown.

In Discussion, two similar exercises were explored: finding the MLE for µ when σ2 was known, and finding the MLE for σ2 when µ was known. This time, both are unknown, but those Discussion exercises will still be helpful.

(a) Find the MLE’s for both µ and σ2.

Hint: The likelihood and log-likelihood functions for both will be the same. But when you get to Step 3, you’ll get two expressions: one from taking the derivative with respect to µ, and the other by taking the derivative with respect to σ2.

(b) Show that the MLE  is a biased estimator for σ2 when µ is unknown. Calculate the value of the bias of .

(c) Based on your answer to part (b), what would you need to do to the MLE to create an unbiased estimator for σ2 when µ is unknown? Show that your new statistic based on the MLE is unbiased.

Hint: What is the name/symbol of that “new statistic”? You have seen it before.