Winter Examination Period 2023 — January — Semester A
应用统计final代写 It is defined by a parameter p and models the probability distribution of the number of Bernoulli trials needed to get one success.
ECS764P APPLIED STATISTICS
Duration: 2 hours
This is a 2-hour online exam, to be completed during a fixed 4-hour window.
You MUST submit your answers before the exam end time.
All instructions and guidelines from the exam page should be followed.
This is an open-book exam and you may refer to lecture material, text books and online resources. The usual referencing and plagiarism rules apply, and you must clearly cite any reference used. 应用统计final代写
Calculators or Python notebooks are permitted in this examination.
Answer ALL questions
You MUST adhere to the word limits, where specified in the questions. Failure to do so will lead to those answers not being marked.
YOU MUST COMPLETE THE EXAM ON YOUR OWN, WITHOUT CONSULTING OTHERS.
Question 1
Consider the density function given by
where K is some constant positive real.
(a)Find the value of K for which the density above defines a probability density function.[6 marks]
(b)Let µ be the probability measure defined by this density.What is the probability mass of the interval 
under µ? In other words, what is µ ()?
(c)Compute the mean and the variance of the probability distribution.[6 marks]
(d)Compute the CDF of the distribution above.[7 marks]
Question 2 应用统计final代写
The Geometric Distribution is a family of distributions on the set N0 of non-zero natural numbers. It is defined by a parameter p and models the probability distribution of the number of Bernoulli trials needed to get one success. In other words how many time a coin needs to be tossed before a head appears. Formally, it is defined by
Geo (p) ({k }) = (1 − p)k−1p
(k − 1 tails followed by one head.)
(a)Compute the following probabilities and express them as a fractions (you don’tneed to numerically evaluate the fractions)
Geo (1/2) ({5}), Geo (3/4) ({4}), Geo (1/3) ({4})[3 marks]
(b)Supposewe’re sampling four times from a geometric distribution with unknown param- eter p and that we observe (4, 6, 5, 4). Compute the probability of these observations
Geo (p) ⊗ Geo (p) ⊗ Geo (p) ⊗ Geo (p) ({(4, 6, 5, 4)})
for p = 1/10, p = 1/4 and p = 1/2. This time you should use a calculator or a notebook to give a numerical answer. Which parameter best explains the observations?[5 marks]
(c)Youare now going to compute the MLE pˆ for the parameter p, based on the observa- tions from the previous question.
(i)You want to find the parameter p which maximises the probability of theobserva- tions. Write down explicitly this probability (as a function of p).
(ii)To find the optimal p we will differentiate this expression r.t. p, but in order to simplifythis calculation we first apply a function to this expression. Which function is it, and what do we gain by applying it?
(iii)Apply this function to the expression you wrote down in (i) and simplify the expression as much as you can.[7 marks]
(d)Solve the MLE optimisation problem for the function of p which you have written in the previous step by setting its derivative r.t. p as zero and solving for p. What is pˆ for the observations (4, 6, 5, 4) given earlier? Consider an arbitrary sequence of samples (x1, … , xn) and generalise what you have just done. What is the formula for pˆ in general?[10 marks]
Question 3 应用统计final代写
(a)Considerthe array [3,1,4,1,5,9]. Compute the sample mean, the sample median and the sample mode (you can express your answers as fractions).[6 marks]
(b)Suppose we add an additional observation x < 3 to this What is the largest value of x for which the mean will be smaller or equal to themedian?[9 marks]
(c)(i)Consider an array of samples of the shape [0, x , 0, x , 0, x , … , 0, x ]. Find the values of x for which the biased sample variance is larger than the range.
(ii) Consider an arbitrary array of samples [x1, … , xn]. Give one condition on these samples which would guarantee that the unbiased sample variance is zero.[10 marks]
Question 4
Consider the probability measure d on the set {2, 3} defined by
d ({2}) = 1 − p, d ({3}) = p
for some p ∈ (0, 1). In other words, d is a Bernoulli distribution shifted by 2 to the right.
(a)(i)Compute the expectation E [d ] of d .
(ii) Compute the variance Var [d ] of d .[7 marks]
(b)(i) What is the support of d + d?
(ii) Use the definition of the pushforward measure to compute 应用统计final代写
(d + d )({4})
from first principles. Justify (briefly) every step of your computation.[8 marks]
(c)You are asked to test the null hypothesis that p =3 with confidence level α = 99%. Having drawn 100 samples independently from d , you observe a sample mean of 2.84. Using the central limit theorem together with the expectation and variance which you have computed above, determine whether you can reject H0 or not. Briefly justify the steps in your computation.
(You may use the fact that αN(0, σ2) = N(0, α2σ2) and α + N(0, σ2) = N(α, σ2) for all α ∈ R. You should use a calculator or notebook for the final numerical computation).[10 marks]
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