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NORMAL DISTRIBUTION • The probability density function – f(x), area and the effect of changing mean and variances • Standard normal distribution – Relationship between x and z – Table, z values, area and probability – Given z find area, use of table and Excel – Given area find z, notation zA, and use of table and Excel • Given x, find probability • Given probability, find x • Normal approximation to binomial and continuity correction 1 NORMAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION • If a random variable X with mean and standard deviation is normally distributed, then its probability density function is given by 1 1/ 2 x / 2 f x e 2 2 NORMAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION f (x ) Mean, =50 SD, =10 0.0500 f (x ) 0.0400 Area between the vertical lines = P (40x 60) 0.0300 Area under the curve = 1.00 0.0200 0.0100 0.0000 0 20 40 60 x - VALUES 80 100 3 NORMAL DISTRIBUTION EFFECT OF CHANGING STANDARD DEVIATION SD,=10 0.0500 SD,=15 f(x) 0.0400 SD,=20 0.0300 0.0200 Mean,= 100 0.0100 0.0000 0 50 100 150 200 4 x-values NORMAL DISTRIBUTION EFFECT OF CHANGING MEAN 0.0500 f(x) 0.0400 0.0300 0.0200 SD,10 0.0100 0.0000 50 100 150 x -values 200 250 5 STANDARD NORMAL DISTRIBUTION 0.4500 Mean=0 SD=1 0.4000 0.3500 f (x ) 0.3000 0.2500 0.2000 0.1500 Area under the curve = 1 0.1000 0.0500 0.0000 -6 -4 -2 0 z -VALUES 2 4 6 6