For this project, I have provided a detailed skeleton of MATLAB code, which is posted along with the assignment. I have also provided the MATLAB function randx for use in Section 3.3. Although you may program in any language you desire, you should look at these for some hints.
2 BACKGROUND
By now, you should know something about histograms; after all, we discussed them in ENES101! Hint: At the very least, take a look at help histogram in MATLAB, even if you are programming in Python or something else. The process we’ll investigate in this lab is shown in Figure 1. Page: 2 S22 CMPE320 Project 1 rev 0A.docx saved 1/31/22 2:05:00 PM printed 1/31/22 2:05:00 PM Figure 1 Histogram to PMF to PDF
Referring to the top box of Figure 1, we take a collection of events from multiple independent trials of a specific experiment, assign numbers to those events by means of a random variable, and then sort the values of the random variable into bins, counting the number in each bin. This creates our “N-point Histogram”. The bin width is and the bin centers are .
Referring to the middle box of Figure 1, if we increase the number of trials in our sample to a very large number, and then divide the number of occurrences in each bin (i.e., the value of the raw histogram in that bin) by the total number of independent trials, we form a naïve estimate of the probability that the value of the random variable falls in that bin. Clearly, if we sum up all of these estimates, we should compute exactly 1.000…, because we will have accounted for all of the trials. Thus, in a simple view, this scaled histogram represents the probability mass function (PMF) of discrete events where the value of our random variable, , “rounds” to the value at the center of the bin, .
Referring to the lowest box, if we then also let the size of the bin, decrease as we make the number of trials larger, the histogram should more and more closely approach the analytical probability density (PMF), where represents the center of the bin. We write this infinitesimally small bin width as , just like in Calc II. The following experiments will illustrate this process. 3 EXPERIMENTS 3.1 PMF for a single fair die Using the MATLAB function randi(imax, m, n)1, model the number of dots showing on a fair six-sided die. In this case the number of dots – that is, resulting random integer – is the random variable. Each element of the returned matrix of values is one trial. Generate histograms using 120; 1200; 12,000; 120,000 trials and generate the unnormalized and normalized histograms where the y-value associated with each bin is . In each case, compute and report the (sample) mean and (sample) variance of your trials. Hint: use the appropriate MATLAB functions for this! Discuss your observations as the number of points increases. How do the histograms vary (or not) from what you expect? Note that your histogram is an estimate of the Probility Mass function, because “number of dots” is a discrete random variable. It doesn’t matter how small you make the bins, you will still get values only at the integers from 1…6. For this problem, the analytical expected value, or mean, is , and the analytical variance 2.9167. How do the (sample) mean and (sample) variance compare with the analytical values? What do you observe as the value of N increases? This section requires four plots, one for each number of trials. You only need plot the normalized histogram. Hint: Look at the skeleton solution
3.2 PMF for binary strings Now generate a series of strings of 100 binary values, where each value can be either 0 or 1. First, let the probability of a value of 1 be . See the hints in the accompanying MATLAB script. Generate a large number of these strings. For this problem, the value of the random variable is the index of the first 1 in the string, not the string itself. This is a mapping from a random event (the string) to a[n] (integer) number and thus, a random variable. For each string of 100 binary values, compute the value of the random variable and then create a histogram of these values. Follow the same process as in 3.1: scale the histograms to compute a PMF for this geometrically distributed random variable. Determine the value of the analytical, or population, PMF based on your value of , and plot the analytical values on the same axis of the scaled histogram or PMF. Compute your sample mean and variance, and the analytical or population, mean and variance, and for each value of and each value of Repeat this for . Then do the entire process over again, including for various values of for . For each of the nine cases (Product Rule! ) answer the following questions. How do the histograms vary (or not) from what you expect? How do the (sample) mean and (sample) variance compare with the analytical values? Hint: A table might be a good way to summarize your answers. This section requires nine plots. MATLAB subplots are recommended.
3.3 PDF for an exponentially distributed random variable. Using the provided MATLAB function randx(n, k, lambda)2, generate histograms for independent trials of , first using a raw (i.e unscaled) histogram and then using the ‘Normalization’,’pdf’ option in the MATLAB function histogram. Plot the resultant normalized histograms of each set of trials. On the same set of axes, plot the values value of the pdf , where is the value at the center of each bin. Go to the text (or lecture slides) and review the definition of the probability density function. In each case, compute the sample mean and variance from your experiments) and the analytical (population) mean and variance. For this problem, the analytical expected value is and the analytical variance is . Answer and discuss these questions in your report: What scale factor creates the normalized histogram from the raw histogram? Discuss this scaling in terms of the two-step process shown in Figure 1. How does that inform your understanding of the meaning of the pdf? Comment on why the scaling was necessary. Hint: The answer “to make it fit” is not acceptable. How do the (sample) mean and (sample) variance compare with the analytical (population) values? What trend do you observe? This section requires three plots, each of which has the normalized histogram and the analytical (population) pdf on the same axes. Each plot represents one set of trials.
3.4 PDF for a unit variance normal or Gaussian distributed random variable.
Using the built-in MATLAB function randn(n,k)3 generate histograms for 10, 1000, and 100,000 independent trials of a zero mean, unit variance, Gaussian (Normal) random variable. Create the raw and normalized histograms as in Section 3.3. Plot the scaled histograms of each set of trials, and, on the same axis, plot the analytical PDF evaluated at the bin centers. For each set of trials, compute the sample mean and variance and the population mean and variance, as in Section 3.3 above. For this problem, the analytical expected value is and the analytical variance is . As in Section 3.3, answer and discuss these questions in your report: What scale factor creates the normalized histogram from the raw histogram? Discuss this scaling in terms of the two-step process shown in Figure 1. How does that inform your understanding of the meaning of the pdf? Comment on why the scaling was necessary. Hint: The answer “to make it fit” is not acceptable. How do the (sample) mean and (sample) variance compare with the analytical (population) values? What trend do you observe? 2 For those programming in a language other than MATLAB, randx(n,k,lambda) creates an array of random values from the distribution . You may look at the MATLAB code for randx to see how to create your own. 3 For those programming in languages other than MATLAB, randn(n, k)generates independent samples from a zero mean, unit variance Gaussian pdf, . This section requires three plots, each of which has the normalized histogram and the theoretical pdf on the same axes. Each plot represents one set of trials.
3.5 PDF for a normal or Gaussian distributed random variable.
Repeat all of 3.4 with a samples from , that is, a normal (Gaussian) random variable with and . Hint: You will have to modify the output of the MATLAB function randn(n,k)to get the desired pdf. For this problem, the analytical expected value is and the analytical variance is . As in Section 3.4, answer and discuss these questions in your report: What scale factor creates the normalized histogram from the raw histogram? Discuss this scaling in terms of the two-step process shown in Figure 1. How does that inform your understanding of the meaning of the pdf? Comment on why the scaling was necessary. Hint: The answer “to make it fit” is not acceptable. How do the (sample) mean and (sample) variance compare with the analytical (population) values? What trend do you observe? This section requires three plots.
3.6 Computing probabilities from the pdf
Using the unscaled histogram from Section 3.5, count the number of trials that fall between and . Scale this to be a probability by dividing by the total number of trials. Then use your normalized histogram, which models the probability density function, to compute the sample probability that the random variable falls between 1.0 and 3.0. Finally, numerically integrate the true probability density function to find the probability that . Compare your results and discuss any differences. How might the width of the bins affect your answer? There are no plots required in Section 3.6.
