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 customers come in within a time period of length ?

customers in a period of length k T

Idea: Approximate the scenario by a binomial model

Divide into 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

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

0 T

( ) (1 )

( ) lim (1 )

k n k

k n k

n

nP X k p p

k

nP X k p p

k

Theorem 1

( ) lim (1 )

!

k n k

n

k

T

nP X k p p

k

Te

k

Why?

Calculus Formula:

lim 1

lim 1 lim 1

nx

n

n nT

n n

xe

n

T

ne

T

n

lim (1 )

!

k

Tk n k

n

n Tp p

k ke

Why?

(1 )k n knp p

k

lim 1n

T

n

Te

n

lim (1 )

!

k

Tk n k

n

n Tp p

k ke

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

Team HW#1

Team Homework #1

Use the definition of expected value and the Taylor expansion

Do not use the moment generating function.

0 !

kx

k

xe

k

Poisson Distribution

Model arrival process Approximate binomial dist. when is large

(1 ) Vs

!

k

k n k Tn Tp p e

k k

Team Homework #2

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

>fun:=proc(x) #Program to compute the given # piecewise defined function local value; #Function value if x<0 then value:=x^2 fi; #Case for x<0 if x>=0 then value:=x fi; #Case for x>=0 print(value); #Output function value end:

Example 1 Version 12 0

( )0

x xf x

x x

>fun:=proc(x) #Program to compute the given # piecewise defined function local value; #Function value if x<0 then value:=x^2 fi; #Case for x<0 if x>=0 then value:=x fi; #Case for x>=0 print(value); #Output function value end:

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

Structure of the if-blockif condition then

block of statements to be executed

fi;

Statements executed only if the condition is met. Otherwise, the statements will be skipped:

Example 1 Version 22 0

( )0

x xf x

x x

> fun:=proc(x) #Program to compute the given # piecewise defined function local value; #Function value if x<0 then value:=x^2; #Case for x<0 else value:=x; #Otherwise fi; print(value); #Output function value end:

Example 1 Version 22 0

( )0

x xf x

x x

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

> fun:=proc(x) #Program to compute the given # piecewise defined function local value; #Function value if x<0 then value:=x^2; #Case for x<0 else value:=x; #Otherwise fi; print(value); #Output function value end:

Structure of the if-blockif condition then

Statement block 1

else Statement block 2

fi;

There are two cases separated by one condition: the condition is met or else:

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)

2

2

2

1 2 if 0

( ) 2 1 if 0 2

5 if 2x

x x

f x x x

e x

Structure of the if-block

if condition 1 then Statement block 1

elif condition 2 then Statement block 2

... ... elif condition n then

Statement block n else

Final statement block fi;

Homework

Read 1.6 for formatting with printf See webpage