#3 Solution DSO 516: Probability and Data Modeling Professor Amy R. Ward Due: Feb. 10, uploaded to BB, by the beginning of class. Question 1. (100 points) You have volunteered to help with Candidate X's campaign. Candidate X has a very long list of people to call, to see if they will commit their vote to him/her (at least, verbally commit). Every day, you have promised to make calls until two callers verbally commit to vote for Candidate X. Based on past experience, you estimate that for 80% of the calls, no verbal commitment is made. What is the expected number of calls you must make every day? Although you may be able to reason out the answer to this question, you must modify the Macro we wrote in class to find the mean of a Geometric distribution to produce your answer. You must hand in your Excel sheet. Session 9: Common Continuous Distributions PROFESSOR AMY R. WARD DSO 516 PROBABILITY AND DATA MODELING Today's Agenda We will discuss two common continuous distributions, the uniform and the normal. We will see why the normal is a distribution that very commonly arises in practice. The underlying theory is the law of large numbers and the central limit theorem. Uniform Let be a subset of the real line. Suppose that the random variable is equally likely to lie anywhere in the subset. Recall the \"Spinner\" example. The spinner was equally likely to land anywhere in [0,1]; so Spinner pdf: if Spinner cdf: if You can think of as modeling ignorance: There is no reason to assign higher frequencies of occurrence to some values over other values. What is an example of when this might be the case? (Suppose you arrive to a bus stop.) Uniform(a,b): PDF, CDF, and Summary Statistics Let be a subset of the real line. The pdf is: if . The cdf is: if . What is the mean? What is the median? What is the variance? What happens with 2 spinners? Let be the outcome of a uniform(-0.5,0.5). (We spin the spinner, and subtract 0.5, so that the mean is 0.) Let be the outcome of an independent uniform(-0.5,0.5). (We again spin the spinner, and subtract 0.5.) What should be the density function for ? Will it still be uniform? What is the minimum and maximum value can assume? What should be the density function for the average ? Will it still be uniform? What is the minimum and maximum value can assume? What is the expectation of the average? What is the variance? Let's use Excel to gain some insight. Observations The values are becoming more concentrated about the mean (0). In-Class Exercise What is the pdf for 10 spinners? Use Excel. 10 Spinner Histogram Plot The values become even more concentrated about their mean. The pdf looks like a bell-shaped curve. You have just discovered the Law of Large Numbers and the Central Limit Theorem. Law of Large Numbers Let be independent, identically distributed random variables. Example: The spinner is spun times. The average of the random variables is converging to a deterministic (non-random) number that equals the mean of the distribution. This is a random variable. This is a deterministic (non-random) number. This explains why the sample average is so important! Central Limit Theorem (CLT) Let be independent, identically distributed random variables. Example: The spinner is spun times. Consider the random variable What is its mean? What is its variance? The CLT tells us the reason our histogram plot was a bell-shaped curve. The CLT is This is a standard normal random variable. As a consequence, the sum is approximately normally distributed, with mean and variance . This explains why the normal distribution, which we are about to see, is so important! Can you give some examples where the random event of interest can be represented as a sum? Normal Let be a normal random variable with mean and standard deviation (variance) . The pdf is: . Can you see why the mean of this pdf should be ? The variance, controls the spread. The mean, controls where the center is. The standard normal has mean 0 and variance 1. Normal Let be a normal random variable with mean and standard deviation (variance) . The pdf is: . The standard normal has mean 0 and variance 1. Normal: Excel Commands Let be a normal random variable with mean and standard deviation , with pdf and cdf . =Norm.dist() =Norm.dist() If and , so that we have a standard normal, then Norm.s.dist(z,1) Norm.s.dist(z,0) What if you would like to know the value of such that , for a given probability ? Norm.inv() Norm.inv() for a standard normal. Some practice questions. What is the probability a normal with mean -3 and standard deviation 2 is less than -2? What is the probability a standard normal exceeds 1? Find such that for a standard normal. (The 0.25 th quantile.) What have we learned? The Uniform distribution. The law of large numbers. The central limit theorem. The Normal distribution. Next time: The exponential distribution and Poisson processes