Chapter 4 Continuous Random Variables and Probability Distributions
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
pdf Uniform Distribution (over an interval [a, b]) cdf
Weibull Distribution = “shape parameter” =1 Exponential Distribution = “scale parameter” =1
Generalized Gamma Distribution = =2 4
• “Time-to-Event Analysis” Application to • “Time-to-Failure Analysis” • “Reliability Analysis” • “Survival Analysis” Let X = “Time to Failure” = Prob that Failure occurs before time x. = Prob that Failure occurs after time x. “Reliability Function” R(x) “Survival Function” S(x) continuous, increases from 0 to 1 continuous, decreases from 1 to 0
Recall an argument similar to the memory-less property of the exponential distribution… 0 X No Failure i. e. , “Failure” after x? What is the probability of “Failure” by x+ x, given “No Failure” before x?
Recall an argument similar to the memory-less property of the exponential distribution… 0 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”
Reliability Survival function 0 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 9
hazard function reliability function 10
“bathtub curve” 11
12
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. Basic Properties: 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. Proof: Change variable… Let Proof: Not hard, but lengthy in integral.
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 Find the probability that customers with a certain business follows satisfaction is over 50%. a Beta distribution, having p = 5, q = 4.
Standard Beta Distribution General Beta Distribution
- Slides: 22