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37357 Spring 2019 Page 1 of 5

Lab Week 4

Distribution Theory and MLE

Name: Student Number:

Part 1: Poisson Distribution

Let Y1, …, Yn be independent random variables that follow the Poisson distribution

(1) Write down the likelihood function,

(2) Write down the log-likelihood function.

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(3) Calculate 

(4) Find the maximum likelihood estimator of.

Part 2: Unbiased Estimators

Let Yi = μ + εi, i=1,…,n with independent and identically distributed errors εi such that E[εi] = 0 and

Var[εi] = σ2

.

(5) Find the least squares estimator of μ. 

Part 3: Binomial Distribution

Let Y be a random variable that follows the Binomial distribution

(7) Write down the likelihood function,

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(8) Write down the log-likelihood function,

(10) Find the maximum likelihood estimator of.

Open the Excel sheet “MLE Bin Dist.xlsx” on UTSOnline. This sheet calculates based on n = 30

Bernoulli trials both algebraically (the MLE found in Q10) and numerically (by finding the maximum

likelihood from a list of possibilities). You can re-run the trails by pressing F9.

(11) Leave the “true p” at 0.3 and re-run the trails 10 times. Each time you re-run the trials, write

down . Do to algebraic and numeric estimates of p agree with one another in each re-run?

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(12) Calculate the mean and standard deviation of the 10 estimates of p in the previous question.

What do you think will happen to the mean and standard deviation when n is larger?

(13) Increase n from 30 to 60 by creating 30 more trials in Row 7. You can do this by copying the

contents of row 6 from column E to AH. Check that the calculation of n in the top left corner has

changed. Repeat question 11 with the increased sample size.

(14) Increase n from 60 to 90 by creating 30 more trials in Row 8. Repeat question 11 with the

increased sample size.

(15) Calculate the mean and standard deviation of the estimates of p in each of Q13 and Q14.

Comment on the effect of increasing n.

More fun things to play with: change the value of “true p”.