Probability Distributions

Special Distributions

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

Continuous Random Variables Uniform R.V. - uniform on the interval [0, u]

- p.d.f.

- c.d.f.

uxux

xxf uX

if00 if

0 if01

uxux

xxF u

xX

if10 if

0 if0

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

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 575400 XP

Continuous Random Variables Two ways to solve:

(1)

(2) Find area under p.d.f.

0875.0

400575575400

2000400

2000575

XX FFXP

Continuous Random Variables Area under p.d.f.

Rectangle

A = l w

Ans.

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

-500 0 500 1000 1500 2000 2500

0875.017520001

Continuous Random Variables Exponential R.V.

- p.d.f.

- c.d.f.

0 if0 if0

/1 xex

xf xX

0 if10 if0

/ xex

xF xX

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

0

1

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 )

4.17

Continuous Random Variables Soln. We are trying to find

Be careful about parenthesis

2418 XP

1037.0644590.0748248.011

182424184.17/184.17/24

ee

FFXP XX

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:

4/41 x

X exf xX exf 25.025.0

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 .

2u

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

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

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

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)

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

iWE

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

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

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

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

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,

22 QC 22 RD 33 RD 33 QC

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

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

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?

Continuous Random Variables Focus on the Project: (What to do)

Answer all related homework questions relating to your project