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ECE 514 project: Part 1

1 Simulating Random Variables

1. Simulate Tj samples of random variables using MATLAB built-in functions as well as the rejection method, for {Xi}i=1,...,Tj , T1 = 100, T2 = 1000, T3 = 10000 with a PDF that is:

(a) Normal with mean=2 and variance=2.

(b) Uniform on [2, 4].

(c) Exponential with the parameter, λ = 2.

2. For each of the above cases, namely (a), (b) and (c), and also further for each of the observa-tion lengths (i.e. Tj ’s), compute histograms using bin-sizes 3,1,0.5 and 0.25. Also estimate the parameters of each of the populations for each of the Tj ’s.

3. Compare these empirical/computed parameters for each of the populations, to the theoretical ones. How do they compare?

4. If they are somewhat different, can you explain these differences? What are the reasons for these differences?

2 Transforming Random Variables

1. Define  for the THREE different distributions of Xi in Part 1, and compute the associated histograms for {Yj}j=1,2,3 for each Tj .

2. By consulting standard probability density functions (PDF), find the closest PDF which matches each of the histograms for each of the Tj .

3. How does the matching vary with Tj? How can you explain the variation?

3 Convergence of Random Variables

Let M be the mean of Xi’s as described in preceding parts. Following the paper ”Understanding Convergence Concepts: A Visual-Minded and Graphical Simulation-Based Approach”, establish a demo by GUI in MATLAB to show and answer the following questions based on Yj in Part 2:

Reference: Matlab GUI tutorial