MATH2801 Assignment PDF

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MATH2801: Theory of Statistics Assignment Please follow the instructions below for the assignment (worth 15% of the final mark): Due date: Before 12 pm Wednesday 5th August (Week 10). Submission details: 1. This assignment may be completed in a group (max. 4 people), or individually.

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2. There are two Turnitin submission links in Moodle. Each person in the group must submit an electronic copy of the assignment via the INDIVIDUAL Turnitin link. One person must also submit the assignment via the GRADED Turnitin link.

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3. The assignment may be typed or handwritten then scanned. It must be submitted as a pdf file. Please make sure all the names and zIDs of each group member are written on all pages.

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4. Make sure you show all workings. You do not need to include RStudio commands/code. Try to aim for clarity and conciseness! Length: At most 7 pages are allowed for your assignment solutions. This cover sheet must be submitted with your assignment, but it is not counted in the 7-page limit. A single pdf and Word

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version of the coversheet is available in Moodle. Please do not exceed the 7-page limit! Declaration: You must sign and date your submitted assignment, and include the PRINTED names/zIDs of any other group members below:

Your name/zID:

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Group member’s name/zID: Group member’s name/zID: Group member’s name/zID:

I (We) declare that this assessment item is my (our) own work, except where acknowledged, and has not been submitted for academic credit elsewhere, and acknowledge that the assessor

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of this item may, for the purpose of assessing this item: Reproduce this assessment item and provide a copy to another member of the University; and/or, Communicate a copy of this assessment item to a plagiarism checking service (which may then retain a copy of the assessment item on its database for the purpose of future plagiarism checking).

I (We) certify that I (We) have read and understood the University Rules in respect of Student Academic Misconduct. Signed:

Date:

1. This question requires the use of RStudio but you do not need to include the RStudio commands/code used for this question. Make sure to include all plots in your assignment. This part of the question is about the random variable X ∼ Exponential(3/4). (a) Use RStudio to simulate a sample of size n = 200 from the Exponential(3/4) distribution and create a frequency histogram of the simulated data.

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Is the distribution of the simulated data skewed or symmetric? (b) Use RStudio to simulate 1000 samples of size n = 200 from Exponential(3/4). Then, compute the sample means of these 1000 samples, and use these to create a frequency histogram of these sample means.

(c) Construct a normal quantile plot of these 1000 sample means (from part b).

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(d) By making references to the produced plots, explain whether the Central Limit Theorem is verified in this simulation study.

fX (x; θ) = (θ + 1)xθ ,

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2. Let x1 , . . . , xn be a random sample of observations from the model 0 < x < 1, θ > 1.

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(a) Find the method-of-moments estimator θe of θ. (b) i. Find the maximum likelihood estimator b θ of θ. b ii. Compute the Fisher Information of θ. b iii. Hence, compute the standard error of θ.

(c) Obtain the probability density function of Y = − ln(X ).

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3. Let X have a geometric distribution with parameter p. (a) Derive the cumulative distribution function of X. Make sure you show all workings.

(b) Prove that X has the lack of memory property, that is, P(X > k + j | X > k) = P(X > j)

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where k and j are positive integers. Make sure you show all workings. (c) Let X1 , . . . , Xn be a random sample of size n from a geometric distribution with parameter p = 0.6. Find the moment generating function of Y = X1 + · · · + Xn . Make sure you show all workings.

4. Consider a mixture of two variables Y1 and Y2 which has density function fY (y ) = πfY1 (y ) + (1 − π)fY2 (y)

where 0 < π < 1. The parameter π is known as the “mixing proportion”. (a) State the two properties of a density function. (b) Show that since fY1 (y) and fY2 (y) are density functions, so is fY (y). Make sure you show all workings.

5. Let X1 , X2 , . . . , Xn be a random sample (iid) from a random variable X with mean µ and variance σ 2 < ∞. The usual estimator for µ is X n = 1 Pn Xi . i=1

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Assume n > 3. A researcher investigates an alternative estimator for µ, by ignoring Xn−1 and Xn and multiplying X1 by 3, giving ˜n = X

3X1 +

Pn−2 i=2

n

Xi

=

3X1 + X2 + · · · + Xn−2 . n

(a) Show that X˜n is an unbiased estimator of µ. ˜ n ). (b) Determine the mean square error, MSE( X ˜ n ) = 0. (c) Show that lim MSE( X n→∞

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˜ n and X n are unbiased estimators for µ for which (d) Both X

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The researcher must decide which estimator of µ to use.

˜ n ) = lim MSE(X n ) = 0. lim MSE( X

n→∞

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n→∞

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Explain, using concepts learned in MATH2801 which of X˜n and X n you would consider to be a better estimate of µ....