1. 1 1 Slide Continuous Probability Distributions n The Uniform Distribution  a b   n The Normal Distribution n The Exponential Distribution

2. 2 2 Slide  a b The Uniform Probability Distributions The Uniform Probability Distributions  a b  a b x1x1 x2x2 x1x1 x1x1 P(x 1 ≤ x≤ x 2 ) P(x≤ x 1 ) P(x≥ x 1 ) P(x≥ x 1 )= 1- P(x<x 1 )

3. 3 3 Slide The Uniform Probability Distribution n Uniform Probability Density Function f ( x ) = 1/( b - a ) for a < x < b f ( x ) = 1/( b - a ) for a < x < b = 0 elsewhere = 0 elsewherewhere 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=x 1 ) = 0

4. 4 4 Slide The Normal Probability Density Function where  = mean  = mean  = standard deviation  = standard deviation  = 3.14159  = 3.14159 e = 2.71828 e = 2.71828

5. 5 5 Slide The Normal Probability Distribution n Graph of the Normal Probability Density Function  x f ( x )

6. 6 6 Slide The Standard Normal Probability Density Function where  = 0  = 0  = 1  = 1  = 3.14159  = 3.14159 e = 2.71828 e = 2.71828

7. 7 7 Slide The table will give this probability Given any positive value for z, the table will give us the following 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.

8. 8 8 Slide Given z =.83 find the probability

9. 9 9 Slide The Exponential Probability Distribution n Exponential Probability Density Function for x > 0,  > 0 for x > 0,  > 0 where  = mean e = 2.71828 e = 2.71828 n Cumulative Exponential Distribution Function where x 0 = some specific value of x

10. 10 Slide 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 Example

11. 11 Slide Example: Al’s Carwash n Graph of the Probability Density Function x x F ( x ).1.3.4.2 1 2 3 4 5 6 7 8 9 10 P ( x < 2) = area =.4866