概率代考 Probability代写 Midterm代写 考试助攻
982CAS MA 581: Probability Midterm 2 概率代考 Note: • No cheat sheet, notes, or textbook allowed. You are allowed to use a calculator. • Please start a new page for each problem Note...
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数学统计作业代写 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.
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
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)1−y 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)
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)
Let Y1, Y2 be an iid sample of size n = 2 from the pdf
f(y| θ) = 2yθ2 0 < y < 1/θ
Find the value c so that the statistic c(Y1 + 2Y2) is unbiased for 1/θ. (1.0 point)
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)
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)
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)
#set the seed so we all get the same answers
set.seed(123)
#this command generates 1,000 multinomial random variables
simdat <- rmultinom(1000,size=1029,prob=c(0.331,0.489,0.180))
#this is a function to calculate the MLE
thfun <- function(y) {
(2*y[3] + y[2])/(2*1029) }
#this command applies the function to the columns of the simulated data
theta <- apply(simdat,2,thfun)
#makes a histogram, calculates mean and standard error
hist(theta,freq=F,col="lightblue"); mean(theta); sd(theta)
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CAS MA 581: Probability Midterm 2 概率代考 Note: • No cheat sheet, notes, or textbook allowed. You are allowed to use a calculator. • Please start a new page for each problem Note...
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