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Unformatted text preview: Chapter 6: Some Continuous Probability Distributions 77 CHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS ( But well only cover one of them ) In this chapter we introduce a continuous probability distribution that will play a critical role in our study of inferential statistics: the normal distribution. The density of a normal random variable X , with mean and variance 2 , is ( 29 ( 29 2 2 2 1 2 1 -- = x e x f , for + < < - x . Some naturally-occurring data sets are described quite well by a normal distribution: Heights of adults IQ scores Physical characteristics of manufactured items Measurement error Note that a normal curve is entirely described by its mean and its variance. The mean provides the location of the distribution, and the variance determines its spread: Characteristics of the normal distribution: Bell-shaped Symmetric about the mean, median, and mode are equal Standard deviation is Points of inflection at = x Chapter 6: Some Continuous Probability Distributions 78 As with any other continuous distribution, probabilities are found through integration: ( 29 ( 29 ( 29 dx e dx x f b X a P b a x b a -- = = < < 2 2 2 1 2 1 However, because these integrals are difficult to compute, we use tables (or computers) to determine probabilities. Since there are infinitely many combinations of means and variances, one might think we need infinitely many tables. However, all normal distributions can be easily transformed to the standard normal distribution , which has a mean of 0 and standard deviation of 1. Standardizing our x values through the transformation: - = x z , we obtain a standard normal random variable Z . Probabilities are therefore calculated as: ( 29 ( 29 ( 29 ( 29 ( 29...
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