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Lecture 12

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Jeff Racine

Lecture 12 The Normal Distribution A. De Moivre (1667 – 1745) first used this distribution in the year 1733. It is also called the Gaussian Distribution in honour of C. F. Gauss (1777 – 1855) The normal distribution is given by −1 x−μ 2 1 2( σ f (x)= 2×e √2πσ The following are properties of the normal distribution • The normal distribution is symmetric • The curve extends from – inf to inf • The limit of the area under the curve is 1 • The curve is always above the x axis • The mean, median, and mode are equal 2 • When a random variable is normally distributed, we say that X ~ N (μ,σ ) Some measurements of natural phenomena (ranging from people’s heights, weights, or IQs) exhibit frequency distributions that closely resemble the normal distribution The Standard Normal Distribution A random variable is said to have a ‘standard’ normal distribution if it has a normal σ =1 distribution with a mean µ = 0 and a variance , i.e. Z ~ N(0,1) Why is this of interest? Simply because when we want to know areas under any normal distribution, all we need is the area under the standard normal distribution and know how to convert from any normal to the standard normal 2 Suppose that X ~ N μ,σ ) (n
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