## Description

• For the sake of effective learning, if you submit the solution to the assignment as a group, then each member of the group agrees to have participated fully (100%) in performing every part of every question in the assignment.

• If you submit the solution to the assignment (by yourself or as a group), then you agree that every line of code and every line in the report is your (or your group’s) own, and isn’t a copied/modified version of any other source online (on the internet) or offline (in electronic form or paper form or any other form).

• Submit your solution to each problem, i.e., (i) the code, (ii) the results, e.g., graphs or other data, and (iii) the report (in Adobe PDF format), for each question, through moodle. Put the code within the folder “code”, the results within the folder “results”, and the report within the folder “report”.

• Submit all code that allows the TAs to regenerate your results, exactly as they appear in the report.

• Submit a single zip file that contains the solutions to all problems in the assignment.

• To get any possible partial credit for the code, ensure that the code is very well documented. To get partial credit for the derivations, include all derivation steps in their full details.

• To avoid non-deterministic results in each program run, and to make the results reproducible during test time, use rng(seed) where seed is a fixed hard-coded integer in your code.

• If the question suggests the use of some function in Matlab, then you can use a corresponding function in other coding frameworks/languages.

• 5 points are reserved for submission in the proper format.

1. (15 points)

For each of the following distributions, do:

(i) plot the probability density function (PDF) based on the analytical expression. The PDF must appear smooth enough and without apparent signs of discretization.

(ii) plot the cumulative distribution function (CDF) using Riemann-sum approximation. The CDF mustappear smooth enough and without apparent signs of discretization.

(iii) use Riemann-sum approximations to compute the approximate variance (if finite) within a tolerance of 0.01 of its true value known analytically.

• [6 points: 1 + 2 + 3]

Laplace PDF (en.wikipedia.org/wiki/Laplace_distribution#Probability_density_function) with location parameter µ := 2 and scale parameter b := 2.

• [6 points: 1 + 2 + 3]

Gumbel PDF (en.wikipedia.org/wiki/Gumbel_distribution) with location parameter µ := 1 and scale parameter β := 2.

• [3 points: 1 + 2]

Cauchy PDF (en.wikipedia.org/wiki/Cauchy_distribution#Probability_density_function) with location parameter x0 := 0 and scale parameter γ := 1.

2. (15 points)

Consider two independent Poisson random variables X and Y , with parameters λX := 3 and λY :=

4.

• [5 points: 3 + 1 + 1]

Define a random variable Z := X + Y , having a probability mass function (PMF) P(Z).

(ii) What will the PMF P(Z) be theoretically/analytically ?

(iii) Show and compare the values for Pb(Z = k) and P(Z = k) for k = 0,1,2,··· ,25.

• [10 points: 6 + 2 + 2]

Implement a Poisson thinning process (as discussed in class) on the random variable Y , where the thinning process uses probability parameter 0.8. Let the thinned random variable be Z.

(ii) What will the PMF P(Z) be theoretically/analytically ?

(iii) Show and compare the values for Pb(Z = k) and P(Z = k) for k = 0,1,2,··· ,25.

3. (30 points)

Simulate N := 104 independent random walkers (as discussed in class) along the real line, each walker starting at the origin, and each walker taking 103 steps each of length 10−3.

• [5 points]

• [5 points]

For the first 103 walkers, plot space-time curves that show the path taken by each walker (as depicted in the class slides). On the graph, draw each path in a different randomly-chosen color for better clarity.

• [10 points]

Submit the well-documented code for all of the above.

• [5 points: 1 + 4]

Consider a random variable X. Consider a dataset that comprises N independent draws (e.g., modeled by X1,··· ,XN) from the distribution of X.

Use the law of large numbers to show that the random variable Mc := (X1+···+XN)/N converges to the true mean M := E[X] as N → ∞.

Prove that the expected value of the random variable tends to the true variance V := Var(X) as N → ∞. It can also be shown that the variance of the random variable Vb tends to zero as N → ∞; the proof is a bit tedious though (see https://mathworld.wolfram.

com/SampleVarianceDistribution.html).

• [1 points]

Report the empirically-computed mean Mc and the empirically-computed variance Vb of the final locations of the random walkers.

• [3 points]

What should the values of the true mean and the true variance be for the random variable that models the final location of the random walker, as function of the step length and the number of steps ?

• [1 points]

Report the error between the empirically-computed mean and the true mean.

Report the error between the empirically-computed variance and the true variance.

4. (25 points)

Consider a continuous random variable X that has an M-shaped probability density function (PDF) PX(·) as follows:

PX(x) := 0 for |x| > 1, and

PX(x) := |x| for x ∈ [−1,1].

Consider independent continuous random variables {Xi : i = 1,2,··· ,∞} with PDFs identical to that of X.

Define random variables , which have associated distributions PYN(·).

• [5 points]

Write code to generate independent draws from PX(·). Your code can use only the uniform randomnumber generator rand() (no other generator). Submit this code.

• [5 points: 2 + 3]

Show plots of (i) the histogram (with 200 bins) and (ii) cumulative distribution function (CDF), both using M := 105 draws from the PDF PX(·).

• [8 points]

Use the code written in the previous sub-question to write code to generate independent draws from PYN(·). Submit this code.

• [7 points: 3 + 4]

Show plots (separately) of histograms using 104 draws from each of the PDFs PYN(·) for N = 2,4,8,16,32,64.

5. (25 points)

Generate a dataset comprising a set of N real numbers drawn from the uniform distribution on [0,1].

Consider various dataset sizes N = 5,10,20,40,60,80,100,500,103,104.

For each N, repeat the following experiment M := 100 times:

(i) first, generate the data, (ii) then, compute the average µ, and b

(iii) finally, measure the error between the computed average µb and the true mean µ as |µb − µtrue|.

• [5 points] Repeat the above question by replacing the uniform distribution by a Gaussian distribution with µ := 0 and σ2 := 1.

• [5 points] Interpret what you see in the graphs. What happens to the distribution of error as N increases ?

• [10 points: 5 + 5] Submit the well-documented code for both uniform and Gaussian cases.

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