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Ch4.3 Normal Distribution A continuous RV X is said to have a normal distribution with and , where and parameters 0 , if the pdf of X is 1 ( x )2 /(2 2 ) f ( x) e 2 x The normal distribution with parameter values 0 and 1 is called a standard normal distribution. The random variable is denoted by Z. The pdf is 1 z2 / 2 f ( z;0,1) e z 2 The cdf is ( z ) P( Z z ) f ( y;0,1)dy Ch4.3 Ch4.3 Standard Normal Cumulative Areas Shaded area = (z ) Standard normal curve z will denote the value on the measurement axis for which the area under the z curve lies to the right of Shaded area P( Z z ) z Ch4.3 Ch4.3 Nonstandard Normal Distributions If X has a normal distribution with mean deviation , then X Z has a standard normal distribution. Ch4.3 and standard Ch4.3 Normal Curve Approximate percentage of area within given standard deviations (empirical rule). 99.7% 95% 68% Ch4.3 Ch4.3 Percentiles of an Arbitrary Normal Distribution (100 p)th for (100p)th percentile for normal standard normal Normal Approximation to the Binomial Distribution Let X be a binomial rv based on n trials, each with probability of success p. If the binomial probability histogram is not too skewed, X may be approximated by a normal distribution with np and npq . x 0.5 np P( X x) npq Ch4.3