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Continuous Probability Distributions The Uniform Distribution a b The Normal Distribution The Exponential Distribution Slide 1 The Uniform Probability Distributions a x1 x2 P(x1 ≤ x≤ x2) b a x1 b P(x≤ x1) P(x≥ x1)= 1- P(x<x1) a x1 P(x≥ x1) b Slide 2 The Uniform Probability Distribution Uniform Probability Density Function f (x) = 1/(b - a) for a < x < b = 0 elsewhere where a = smallest value the variable can assume b = largest value the variable can assume The probability of the continuous random variable assuming a specific value is 0. P(x=x1) = 0 Slide 3 The Normal Probability Density Function 1 ( x )2 / 2 2 f ( x) e 2 where = mean = standard deviation = 3.14159 e = 2.71828 Slide 4 The Normal Probability Distribution Graph of the Normal Probability Density Function f (x ) x Slide 5 The Standard Normal Probability Density Function where =0 =1 = 3.14159 e = 2.71828 Slide 6 Given any positive value for z, the table will give us the following probability The table will give this probability Given positive z The probability that we find using the table is the probability of having a standard normal variable between 0 and the given positive z. Slide 7 Given z = .83 find the probability z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 .1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 .2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 .3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 .4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 .5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549 .7 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 Slide 8 The Exponential Probability Distribution Exponential Probability Density Function f ( x) where 1 e x / for x > 0, > 0 = mean e = 2.71828 Cumulative Exponential Distribution Function P( x x0 ) 1 e x0 / where x0 = some specific value of x Slide 9 Example The time between arrivals of cars at Al’s Carwash follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866 Slide 10 Example: Al’s Carwash Graph of the Probability Density Function F (x ) .4 .3 P(x < 2) = area = .4866 .2 .1 x 1 2 3 4 5 6 7 8 9 10 Slide 11