The Normal Distribution Notes

The Normal Distribution Notes - The Normal Distribution or...

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The Normal Distribution . . . or Another Distribution Adam Brandenburger * Shellwyn Weston Version 04/19/15 1 Introduction The normal — also known as Gaussian — distribution is surely the probability distribution most studied in introductory courses in probability and statistics. Figure 1 One reason so much attention is given to the normal distribution is that the Central Limit Theorem tells us that if a quantity is made up of the sum or average of a number of independently varying quantities, where the means and variances of these latter quantities are sufficiently ‘well- behaved,’ then the first quantity will be approximately normally distributed. To state a simple version of the Central Limit Theorem more formally, let X 1 , X 2 , . . . be a sequence of random variables which are independently and identically distributed, with common mean μ and common variance σ 2 . Then, if we form the statistic: n i =1 X i - , * Stern School of Business, Polytechnic School of Engineering, Institute for the Interdisciplinary Study of Decision Making, Center for Data Science, New York University, New York, NY 10012, U.S.A., [email protected], adambrandenburger.com. Stern School of Business, Institute for the Interdisciplinary Study of Decision Making, New York University, New York, NY 10012, U.S.A., [email protected]
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the distribution of this statistic will approach that of the normal distribution with mean 0 and variance 1, as n tends to . 1 Equivalently, we can say that, for large n , the quantity n i =1 X i is approximately equal to plus an error term which is normally distributed with mean 0 and variance 2 . Figure 1 depicts the probability distribution of the total score obtained by tossing a fair die twice (independently). The shape of the graph is a little like the Bell curve exhibited by the normal distribution. See Figure 2. 2 The Central Limit Theorem tells us that the shape of the graph will become closer and closer to the Bell curve as the number of tosses increases. Phenomena which are often assumed (or found) to follow a normal distribution include the heights of members of a biological population and the observational errors in a physical experiment.
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