1. 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

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

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

4. Generalized Gamma Distribution
 =  = 2

5. “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)

6. 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?

7. “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”

8. 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”

9. hazard function
reliability function

10. hazard function
reliability function

11. “bathtub curve”

13. Lognormal Distribution
Example: Suppose

14. Lognormal Distribution
Example: Suppose

15. 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

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

17. Standard Beta Distribution

18. Standard Beta Distribution

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

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

21. 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%.

22. Standard Beta Distribution General Beta Distribution