Chapter 4 Continuous Random Variables and Probability Distributions 4.1 - Probability Density Functions 4.2 - Cumulative Distribution Functions and Expected Values 4.3 - The Normal Distribution 4.4 - The Exponential and Gamma Distributions 4.5 - Other Continuous Distributions 4.6 - Probability Plots

(over an interval [a, b]) Uniform Distribution (over an interval [a, b]) pdf cdf

Exponential Distribution Weibull Distribution  = “shape parameter”  = “scale parameter”  = 1  = 1 Exponential Distribution

Generalized Gamma Distribution  =  = 2

“Time-to-Event Analysis” “Time-to-Failure Analysis” “Reliability Analysis” “Survival Analysis” Application to continuous, increases from 0 to 1 Let X = “Time to Failure” = Prob that Failure occurs before time x. continuous, decreases from 1 to 0 = Prob that Failure occurs after time x. “Reliability Function” R(x) “Survival Function” S(x)

No Failure i.e., “Failure” after x? Recall an argument similar to the memory-less property of the exponential distribution… X No Failure i.e., “Failure” after x? What is the probability of “Failure” by x+  x, given “No Failure” before x?

“hazard rate function” Recall an argument similar to the memory-less property of the exponential distribution… X No Failure i.e., “Failure” after x? What is the probability of “Failure” by x+  x, given “No Failure” before x? Divide both sides by x: Take limit as x  0: “hazard rate function”

No Failure i.e., “Failure” after x? Reliability Survival function function X No Failure i.e., “Failure” after x? What is the probability of “Failure” by x+  x, given “No Failure” before x? “hazard rate function” “failure rate function” measures the instantaneous rate of Failure at time x “cumulative hazard function”

hazard function reliability function

hazard function reliability function

“bathtub curve”

Lognormal Distribution Example: Suppose

Lognormal Distribution Example: Suppose

Standard Beta Distribution In order to understand this, it is first necessary to understand the “Beta Function” Def: For any p, q > 0, Both p and q are shape parameters. At x = 0, this term… is 0 if p > 1 has a singularity if 0 < p < 1. At x = 1, this term… is 0 if q > 1 has a singularity if 0 < q < 1. Basic Properties: Proof: Change variable… Let in integral. Proof: Not hard, but lengthy

Standard Beta Distribution Def: For any p, q > 0,

Standard Beta Distribution

Standard Beta Distribution

Standard Beta Distribution Example: The proportion X of satisfied customers with a certain business follows a Beta distribution, having p = 5, q = 4.

Standard Beta Distribution Example: The proportion X of satisfied customers with a certain business follows a Beta distribution, having p = 5, q = 4.

Standard Beta Distribution Example: The proportion X of satisfied customers with a certain business follows a Beta distribution, having p = 5, q = 4. Find the probability that customer satisfaction is over 50%.

Standard Beta Distribution General Beta Distribution