Transcript Focus on the Project

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.
0 if x  0
1
f X x    u if 0  x  u
0 if x  u

- c.d.f.
0 if x  0
x
FX x    u if 0  x  u
1 if x  u

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
P400  X  575
Continuous Random Variables
 Two ways to solve:
(1) P400  X  575  FX 575  FX 400

575
2000
400
 2000
 0.0875
(2) Find area under p.d.f.
Continuous Random Variables
 Area under p.d.f.
0.0006
Rectangle
0.0005
0.0004
0.0003
0.0002
A=lw
0.0001
0
-500
Ans.
0
500
1
2000
1000
1500
2000
 175  0.0875
2500
Continuous Random Variables
 Exponential R.V.
- p.d.f.
- c.d.f.
 0
f X x    1  x / 
  e
if x  0
 0
FX  x   
x /
1

e

if x  0
if x  0
if x  0
Continuous Random Variables
 Graph of expon. p.d.f.
Graph of expon. c.d.f.
1
0
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   17.4 )
Continuous Random Variables
 Soln. We are trying to find P18  X  24
P18  X  24   FX 24   FX 18

 
 1  e 24 / 17.4  1  e 18 / 17.4
 0.748248  0.644590
 0.1037
 Be careful about parenthesis

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:
f X  x   14 e  x / 4
f X x   0.25e 0.25 x
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
u
2 .
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, E Wi  , gives one measure of the
quality of service
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,
 C2  Q2
D2  R2
C3  Q3
D3  R3
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