BCOR 1020Business Statistics

Lecture 14 – March 4, 2008

Overview

• Chapter 7 – Continuous Distributions– Exponential Distribution– Normal Approximation to the Binomial*

Chapter 7 – Exponential Distribution

Characteristics of the Exponential Distribution:• If events per unit of time follow a Poisson distribution, the

waiting time until the next event follows the Exponential distribution.

• Waiting time until the next event is a continuous variable.

Chapter 7 – Exponential Distribution

Characteristics of the Exponential Distribution:

Probability of waiting more than x Probability of waiting less than x

xexXP )( xexXP 1)(

Chapter 7 – Exponential Distribution

Example: Customer Waiting Time• Between 2P.M. and 4P.M. on Wednesday, patient

insurance inquiries arrive at Blue Choice insurance at a mean rate of 2.2 calls per minute.

• What is the probability of waiting more than 30 seconds (i.e., 0.50 minutes) for the next call?

• Set = 2.2 events/min and x = 0.50 min

• P(X > 0.50) = e–x = e–(2.2)(0.5) = .3329 or 33.29% chance of waiting more than 30 seconds for the next call.

• The mean (expected) time between calls is 1/ = .45 minutes (or 27 seconds).

Chapter 7 – Exponential Distribution

Example: Customer Waiting Time

P(X > 0.50) = 0.3329

ClickersIn Santa Theresa, false alarms are received at the downtown fire station at an average rate of 0.3 per day. If we are modeling the time between false alarms (in days), what it the appropriate distribution to use?

A = Binomial

B = Poisson

C = Normal

D = Exponential

Clickers

In Santa Theresa, false alarms are received at the downtown fire station at an average rate of 0.3 per day. What is the mean time between false alarms (in days)?

A = 0.3

B = 2.1

C = 3.3

D = 5.0

Clickers

In Santa Theresa, false alarms are received at the downtown fire station at an average rate of 0.3 per day. What is the probability that more than a week will elapse between false alarms?

A = 0.1225

B = 0.0001

C = 0.7408

D = 0.8775

Clickers

In Santa Theresa, false alarms are received at the downtown fire station at an average rate of 0.3 per day. What is the probability that two consecutive false alarms will occur within 6 hours?

A = 0.0750

B = 0.0723

C = 0.1653

D = 0.8347

E = 0.9277

Chapter 7 – Exponential Distribution

Inverse Exponential:• If the mean arrival rate is 2.2 calls per minute, we

want the 90th percentile for waiting time (the top 10% of waiting time).

• Find the x-value that defines the upper 10%.

Chapter 7 – Exponential Distribution

Inverse Exponential:• P(X < x) = .90 or P(X > x) = .10• So, e–x = .10• -x = ln(.10)

= -2.302585• x = 2.302585/

= 2.302585/2.2 = 1.0466 min.

• 90% of the calls will arrive within 1.0466 minutes (62.8 seconds).

Clickers

In Santa Theresa, false alarms are received at the downtown fire station at an average rate of 0.3 per day. What is the median time between false alarms?

A = 0.30

B = 2.31

C = 3.33

D = 7.68

Chapter 7 – Exponential Distribution

Mean Time Between Events:

• Exponential waiting times are described as Mean time between events (MTBE) = 1/

• 1/MTBE = = mean events per unit of time• In a hospital, if an event is patient arrivals in an ER, and

the MTBE is 20 minutes, then = 1/20 = 0.05 arrivals per minute (or 3/hour).

Chapter 7 – Normal Approximation to the Binomial*

When is Approximation Needed?• Binomial probabilities are difficult to calculate

when n is large.• Use a normal approximation to the binomial.• As n becomes large, the binomial bars become

more continuous and smooth.

Chapter 7 – Normal Approximation to the Binomial*

When is Approximation Needed?• Rule of thumb: when n > 10 and n(1-) > 10,

then it is appropriate to use the normal approximation to the binomial.

• In this case, the binomial mean and standard deviation will be equal to the normal and , respectively.

= n

= n(1-)

Chapter 7 – Normal Approximation to the Binomial*

16 n

Example Coin Flips:• If we were to flip a coin n = 32 times and = .50, are

the requirements for a normal approximation to the binomial met?

• Are n > 5 and n(1-) > 10?

• n = 32 x .50 = 16n(1-) = 32 x (1 - .50) = 16

• So, a normal approximation can be used with …

828.28)1( n