MAT 4830Mathematical Modeling

Section 1.3

Conditional Statements

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Questions

What is the purpose of a conditional statement?

Questions

Describe a conditional statement in Maple.

Preview

Review Poisson Distribution Introduces the conditional statements Allow the flow of control to branch into

two or more sections of codes based on the truth values of a control expressions

Example 0

On average, random customers per hour come into a local Starbucks during the morning rush hours.

customers per hour

Example 0

What is the probability that exactly k customers come in within a time period of length T?

customers in a period of length k T

Idea: Approximate the scenario by a binomial model

Divide T into n subintervals with equal length. Each interval is small enough such that only at most one customer comes in within the subinterval.

0 T

Consider this as a binomial model.

(a customer walks in within a subinterval) ?p P

Idea: Approximate by a binomial model

r.v. X=no. of customers comes in within a

time period of length T

0 T

( ) (1 )

lim (1 )

k n k

k n k

n

nP X k p p

k

np p

k

Idea: Approximate by a binomial model

r.v. X=no. of customers comes in within a

time period of length T

0 T

( ) (1 )

lim (1 )

k n k

k n k

n

nP X k p p

k

np p

k

Theorem 1

( ) lim (1 )

!

k n k

n

k

T

nP X k p p

k

Te

k

Proof of Theorem 1

Poisson Distribution P(,T)

( , )

Prob. Density Fun. ( ) ( ) , 0,1,...!

Mean

Std. D.

k

T

X P T

Tf k P X k e k

kEX T

T

Expectation: You should be able to prove this without

looking at a reference.

Poisson Distribution

Model arrival process Approximate binomial dist. when n is

large

(1 ) Vs

!

k

k n k Tn Tp p e

k k

Team Homework

A newsboy sells newspapers outside Grand Central Station. He has on average 100 customers per day. He buys papers for 50 cents each, sells them for 75 cents each, but cannot return unsold papers for a refund. How many papers should he buy?

To maximize the expected profit

Zeng Section 1.3

Example 1 Consider the piecewise defined function

2 0( )

0

x xf x

x x

For each interval, we need a different formula to compute the function values

Example 1 Consider the piecewise defined function

2 0( )

0

x xf x

x x

For each interval, we need a different formula to compute the function values

Q: Input=? , Output=?

Example 1 Version 12 0

( )0

x xf x

x x

Example 1 Version 12 0

( )0

x xf x

x x

Structure of the if-block

Example 1 Version 22 0

( )0

x xf x

x x

Example 1 Version 2

> fun(-2);fun(2);42

2 0( )

0

x xf x

x x

Structure of the if-block

Example 2

We need 3 branches

2

2

2

1 2 if 0

( ) 2 1 if 0 2

5 if 2x

x x

f x x x

e x

Example 2

2

2

2

1 2 if 0

( ) 2 1 if 0 2

5 if 2x

x x

f x x x

e x

Example 2

> fun(-3);fun(1);fun(3);-143

5e(-1)

Structure of the if-block

Homework

Read 1.6 for formatting with printf See webpage