# Normal Distribution PDF and its Moment Generating Function

May 5th, 2015
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Texas A & M University
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A walk through of normal distribution, and its PDF is provided. The moment generating function of normal distribution is derived. This is a work in progress, and depending upon response I will add more examples. So far it consists of 10 pages and 8 examples. Feedback is welcome.

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Normal Distribution PDF and its Moment GeneratingFunctionDraft copybyStudent HelperApril 2015c2015Student Helper. All rights reserved.Chapter 1PDF and Moment GeneratingFunctionOne of the most common continous distributions that you will incur in engineering, science and statistics is the normal distribution. The normal distributionis sometimes informally called as the bell curve owing to its bell-shaped look.In this chapter, a brief introduction to the different properties of a normal distribution is provided. In the subsequent chapters, we will look at some of theseproperties in details.1.1Normal PDF and its characteristicsA continuous distribution is usually defined in terms of its probability densityfunction or PDF. Let X be normally distributed random variable, with mean and standard deviation . Then, the PDF, fX (x) of the normally distributedrandom variable X is of the form:1fX (x) = 21 x !2e 2.(1.1)Possibly, the simplest of all normally distributed random variables are theones known as the standard normal variables. For these variables, = 0, and = 1, and the equation 1.1 simplifies to:x21fX (x) = e 2 .21(1.2)0.45 = 0, = 1 = 0, = 2 = 0, = 3 = 1, = 1 = 2, = 2 = 3, = 30.40.350.3PDF0.250.20.150.10.0501086420246810XFigure 1.1: The PDF of normal distributions corresponding to six sets

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