1. Probability Distributions
Special Distributions

2. Continuous Random Variables
Special types of continuous random variables:
Uniform Random Variable
every value has an equally likely chance of occurring
Exponential Random Variable
average time between successive events

3. Continuous Random Variables
Uniform R.V. - uniform on the interval [0, u]
- p.d.f.
- c.d.f.

4. Continuous Random Variables
Graph of uniform p.d.f. Graph of uniform c.d.f.

5. Continuous Random Variables
Ex. Suppose the average income tax refund is uniformly distributed on the interval [$0, $2000]. Determine the probability that a person will receive a refund that is between $400 and $575.
Soln. Note that we are trying to find

6. Continuous Random Variables
Two ways to solve:
(1)
(2) Find area under p.d.f.

7. Continuous Random Variables
Area under p.d.f.
Rectangle
A = l w
Ans.

8. Continuous Random Variables
Exponential R.V.
- p.d.f.
- c.d.f.

9. Continuous Random Variables
Graph of expon. p.d.f. Graph of expon. c.d.f.

10. Continuous Random Variables
For exponential r.v., the value of is the average time between successive events
Ex. Suppose the average time between quizzes is 17.4 calendar days. Determine the probability that a quiz will be given between 18 days and 24 days since the last quiz. (Note: this is and exp. r.v. with )

11. Continuous Random Variables
Soln. We are trying to find
Be careful about parenthesis

12. Continuous Random Variables
Note: Formula for p.d.f. has a fraction that can be written in a decimal form:
Ex. The following formulas are identical:

13. Continuous Random Variables
For an exponential r.v., the mean is ALWAYS equal to .
For a uniform r.v., the mean is ALWAYS equal to

14. Continuous Random Variables
Focus on the Project:
Examining the shape of the graph (histogram) may help us determine information about the type of distribution of a random variable

15. Continuous Random Variables
Focus on the Project:
Let Ab be the time, in minutes, between consecutive arrivals at the 9 a.m. hour on Fridays
Let Au be the time, in minutes, until the first customer arrives at the 9 a.m. hour on Fridays
Ab and Au have the same distribution and we will call the continuous random variable A

16. Continuous Random Variables
Focus on the Project:
Similarly, we will let B be the continuous random variable that is the time, in minutes, between arrivals or until the first arrival of the 9 p.m. hour
We don’t know the distributions of A and B, but the shapes of their histograms leads us to think that the distribution may be exponential

17. Continuous Random Variables
Focus on the Project:
Let S represent the length of time, in minutes, during which a customer uses an ATM
This continuous random variable has an unknown distribution (certainly not exponential)

18. Continuous Random Variables
Focus on the Project:
Suppose we open i ATMs (i = 1, 2, or 3)
Let Wi be the continuous random variable that gives the waiting time, in minutes, between a customer’s arrival and the start of their service during the 9 a.m. hour
The expected value, , gives one measure of the quality of service

19. Continuous Random Variables
Focus on the Project:
Let Qi be the finite random variable that gives the number of people being served, or waiting to be served when a new customer arrives during the 9 a.m. hour
The number of people waiting is a concern for customer satisfaction

20. Continuous Random Variables
Focus on the Project:
Let Ci be the finite random variable that gives the total number of people present when a customer arrives during the 9 a.m. hour
Total number present is a concern for customer satisfation

21. Continuous Random Variables
Focus on the Project:
We define similar variables for the 9 p.m. hour on Fridays:
Let Ui be the continuous random variable that gives the waiting time, in minutes, between a customer’s arrival and the start of their service during the 9 p.m. hour

22. Continuous Random Variables
Focus on the Project:
Let Ri be the finite random variable that gives the number of people being served, or waiting to be served when a new customer arrives during the 9 p.m. hour
Let Di be the finite random variable that gives the total number of people present when a customer arrives during the 9 a.m. hour

23. Continuous Random Variables
Focus on the Project:
If only one ATM is open, C1= Q1 and D1= R1
When two or three ATMs are in service,

24. Continuous Random Variables
Focus on the Project:
Eventually, we will simulate to estimate means and some probabilities for all random variables
We will also find the maximum for the variables

25. Continuous Random Variables
Focus on the Project: (What to do)
Let A, Wi, Qi, and Ci be random variables that are similar to the class project, but apply to your team’s first hour of data
Let B, Ui, Ri, and Di be random variables that are similar to the class project, but apply to your team’s second hour of data

26. Continuous Random Variables
Focus on the Project: (What to do)
Let S be the length of time, in minutes, during which a customer uses an ATM as given in your team’s downloaded data
Which random variables might be exponential?
Which random variables are not exponential?

27. Continuous Random Variables
Focus on the Project: (What to do)
Answer all related homework questions relating to your project