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# 数学统计作业代写 数学作业代写 统计作业代写 数学代写

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## Homework 3

Instructions: Solve the problems in the spaces provided and save as a single PDF. Then upload the PDF to Canvas Assignments by the due date. The recommended procedure is to download and print the homework. Fill in your solutions. Then scan the document and upload to Canvas Assignments. If this is not feasible, you may solve the problems on your paper, scan your solutions, then upload to Canvas. Neatness and presentation are important. Late homework not accepted. Show all work. Total points: 30

### 1) 数学统计作业代写

Let Y1, Y2, . . . , Yn be independent and identically distributed random variables from the Bernoulli distribution with parameter 0 < p < 1.

f(y|p) = py (1 p)1y    y = 0, 1

a) Find the method of moments estimator for p. (0.5 points)

b) Find the MLE (maximum likelihood estimator) for p. (1.5 points)

### 2)

Let Y1, Y2, . . . , Yn be independent and identically distributed from the distribution with density

f(y| θ) = θ cθ y(θ+1)             y > c.

where c > 0 is a constant and θ > 0. Find the MLE for θ. (1.5 points)

### 3) 数学统计作业代写

Let Y1, Y2 be an iid sample of size n = 2 from the pdf

f(y| θ) = 22             0 < y < 1

Find the value c so that the statistic c(Y1 + 2Y2) is unbiased for 1. (1.0 point)

### 4)

Let Y1, Y2, . . . Yn be iid from the normal density N(µ, σ2 ) and assume that σ2 is known.

a) What is the Fisher information I1(µ)? (1.5 points)

e) Let Y = max(X1, . . . , Xn) = X(n). Find the bias (2.0 points)

bY (θ) = E[Y ] θ.

f) How can you correct Y to make it unbiased? (1.0 point)

c) Based on b) what is an approximate 95% confidence interval for θ? (0.5 points)

### 8) 数学统计作业代写

Let Y1, Y2, . . . , Yn be independent and identically distributed from the geometric distribution with parameter 0 < θ < 1,

f(y| θ) = θ (1 − θ)y−1   y = 1, 2, 3, . . .

c) Find an approximate 95% confidence interval for θ based on part b). (0.5 points)

### 10) Let Y ∼ N(θ, 1). 数学统计作业代写

a) Is this distribution a member of the exponential family? (1.5 points)

b) Now let Y1, . . . , Yn be iid N(θ, 1). Find a sufficient statistic for θ. (1.0 point)

### 11) Introduction to the Bootstrap.

`#set the seed so we all get the same answersset.seed(123)#this command generates 1,000 multinomial random variablessimdat <- rmultinom(1000,size=1029,prob=c(0.331,0.489,0.180))#this is a function to calculate the MLEthfun <- function(y) {  (2*y + y)/(2*1029) }#this command applies the function to the columns of the simulated datatheta <- apply(simdat,2,thfun)#makes a histogram, calculates mean and standard errorhist(theta,freq=F,col="lightblue"); mean(theta); sd(theta)`

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