Chapter 12 Continuous Random Variables and their Probability Distributions

Probability Distributions of a Continuous Random Variable For a continuous random variable X, a probability density function such that

Probability Distributions of a Continuous Random Variable For a continuous random variable X, a probability cumulative function:

Mean & Standard Deviation of a Continuous Random Variable

Continuous Probability Distributions Continuous Uniform Distribution Normal Distribution Exponential Distribution Erlang and Gamma Distributions Weibull Distribution Lognormal Distribution Beta Distribution

Continuous Uniform Distribution Probability Density Function Mean Variance

Normal Distribution Probability Density Function, with parameter , where -  0 Mean Variance

Normal Distribution The curve is symmetric about the mean The mean, median, and mode are equal The tails of the curve extend indefinitely

Standard Normal Distribution A normal random variable with parameter  =0, and  =1 Cumulative Distribution: Table II in Appendix A Convert x to z

Standard Normal Distribution -6  -5  -4  -3  -2  -1  01  2  3  4  5  6  

Exponential Distribution Probability Density Function, with mean, where >0, and x>0 Cumulative probability Mean Variance

Exponential Distribution

Exponential Distribution Alternate Definition Probability Density Function, with rate, where >0, and x  0 Cumulative probability Mean Variance

Exponential Distribution Example An HR department wishes to study the need for hiring new secretaries. It is estimated that the amount of time that a secretary stays in the job can be described as an exponential distribution with a mean of 26 months. The company just hired a new secretary. Calculate the probabilities of the following events: The secretary has to be replaced within the first year. The secretary has to be replaced during the third year. The secretary remains in the position for more than 5 years