Statistics 312 15 Normal Distribution 1 The Normal Distribution One important group of density curves are normal bell curves A RV X has a normal distribution with parameters mean and standard deviation 0 if the pdf of X is f x 1 e 2 2 1 x 2 x Standard Normal RV A standard normal RV denoted Z is a normal RV with 0 and 1 The pdf is f z 0 1 1 z e2 x 2 2 Statistics 312 15 Normal Distribution 2 Note 1 For a normal curve approximately 68 95 99 7 of the observations are within 1 2 3 standard deviations of the mean respectively 2 If X is a normal random variable with mean and variance then a standard X m normal random variable is obtained by Z s Ex P 0 z 2 34 4904 P z 2 34 5 4904 9904 P z 2 34 5 4904 0096 P 1 11 z 2 34 4904 3665 1239 P 1 11 z 2 34 4904 3665 8569 Find Z is such that P z Z 0 80 i e find the 80th percentile P z 0 84 2995 P z 0 85 3023 Therefore Z 84 Statistics 312 15 Normal Distribution 3 Find Z is such that P z Z 0 15 P z 1 03 P z 1 03 5 3485 1515 P z 1 04 P z 1 04 5 3508 1492 Z 1 04 Ex Suppose X is a normal RV with 20 and 10 Using z P x 35 P z 35 20 10 P z 1 50 5 4332 9332 x Statistics 312 15 Normal Distribution Conversely we can convert back from standard scores to the original units by using X Z Find X is such that P x X 60 P z 0 25 5 0987 5987 P z 0 26 5 1026 6026 X 20 25 10 22 50 Note Bad formatting above 2022 5 represents 20 and X 22 5 Read pp 184 194 Prob 5 6 5 8 5 11 5 13 5 14 4
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