Description

c Prof. R.S. Sreenivas
1 Introduction
We have an unknown, possibly unfair, three-sided dice (cf. figure 1). You get one of three outcomes (say, xi ∈{1,2,3}) when you roll this dice. We have no clue as to what Prob(xi = 1),Prob(xi = 2), and Prob(xi = 3) are. All we have to go on is that Prob(xi = 1)+Prob(xi = 2)+Prob(xi = 3) = 1. We also know that by repeated tosses of this dice we can generate a string of i.i.d. outcomes
. Empirically estimating these probabilities are ruled-out, as there is no way to estimate the individual probabilities with certainty using a finite-set of outcomes.
We are asked to use this unknown, possibly unfair, three-sided dice as the only available source of “randomness,” and to (a) generate an i.i.d. sequence of exponential RVs with rate λ (to be input by the user), and (b) to generate continuous-RVs that are from a unit-normal distribution. That is, we have to generate a set of Univariate Gaussian RVs such that yi ∼ N(0,1).

Figure 1: Three-sided-dice.
2 Discussion
We know that if we have a way of generating u.i.i.d. RVs we can (a) generate the Exponential RVs using the Inverse Transform Method, and (b) generate the Unit-Normal RVs using the Box-Muller Transform. We are not permitted to use get uniform() for this exercise. All we have is the unknown, possibly unfair, three-sided dice. Essentially, we have to find a way of “converting” the outcomes of repeated-tosses of this three-sided dice into u.i.i.d. RVs.
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I will go over the process of simulating a fair-coin (i.e. simulating a coin whose probability of “heads” and “tails” being equal) using the unknown, possibly unfair, three-sided dice, in class. This method is an illustration of a largeclass of “accept/reject” algorithms that find use in different forms-and-shapes in data analytics. The resulting simulated-fair-coin is used to generate u.i.i.d. RVs. Following this, the rest of the programming exercise should be straightforward.
3 Requirements
What I need from you:
(a) an exponential RVs generator with a user defined rate (i.e. λ; FYI, the mean of exponential-RV distribution is (b) an unit-normal RV generator.
This code should take the following inputs at the command line – #trials (for CDF generation); the rate of the exponential process, the name of the output file for the unit-normal CDF, and the name of the output file for the exponential CDF. The intent to compare the theoretical- and empirical-CDF plots to verify the correctness of your code. If you are lost, watch the video instructions on Compass.
I have provided a hint on Compass for anyone that needs it. If you are working off the hint, you essentially have to fill-in the requisite C++ code in the commented spots where it says
“// write code here.”
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