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#1 2013-03-07 14:59:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Simulations

Hi;

This problem appears in the Devore book "Probability and Statistics" and has various answers on the internet.

The weekly demand for propane gas (in thousands of gallons) from a particular facility is a random variable X with PDF

What is the expected usage for gas for the coming week?

A number of different answers are given. How can we get the answer using a CAS and simulating the process?

1) Define the PDF in mathematica as:

myPDF[]=ProbabilityDistribution[2(1-1/x^2),{x,1,2}];

2) Run the simulation a million times.

data=RandomVariate[myPDF[],10^6];

Below is a histogram of data.

3) Get the expected value:

Mean[data]

1.61357 is returned. So the facility expects to use about 1613 gallons of propane this week.

View Image: 2013-03-07_184646.gif

Last edited by bobbym (2013-03-07 15:35:57)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#2 2013-03-07 21:04:34

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

And it agrees with my answer of 3-2ln(2).


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2013-03-07 21:12:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

More importantly were you able to duplicate the answer with your package?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#4 2013-03-07 21:14:32

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

I will try later.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2013-03-07 21:17:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

Okay, you will find it interesting.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#6 2013-03-07 21:57:29

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

What kind of distribution would that be?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#7 2013-03-07 22:08:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

The one given in post #1 of course!


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#8 2013-03-07 22:11:16

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Yes, but what kind is it? Maxima needs me to specify if it is a normal, geometrical, exponential,...


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2013-03-07 22:36:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

It is none of them.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#10 2013-03-07 22:41:35

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

I am not sure how to do it, then.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#11 2013-03-07 22:43:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

Would you like one way how?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#12 2013-03-07 22:45:55

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Yes.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#13 2013-03-07 22:49:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

Hi;

I am playing with a method right now but do not have all the kinks ironed out of it. I will post what I have.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#14 2013-03-07 22:50:20

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

That is okay.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#15 2013-03-07 23:02:23

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

I want to draw random numbers from the PDF y = 3x^2 from 0 to 1.

So I solve for x

Follow?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#16 2013-03-07 23:02:56

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Yes.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#17 2013-03-07 23:08:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

Then I take the RHS of that equation where I solved for x and integrate wrt y.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#18 2013-03-07 23:12:27

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Okay. What now?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#19 2013-03-07 23:13:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

I replace each y in the answer with a random variable that is uniformly distributed between 0 and 1.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#20 2013-03-07 23:31:29

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Hm, why does it work?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#21 2013-03-07 23:33:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

It doesn't, I think I have it now though. To follow you must know how to generate random numbers between a and b in maxima. Do you follow?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#22 2013-03-07 23:36:42

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Yes. I have the function.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#23 2013-03-07 23:38:30

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

Okay a little theory before we go any further. If you generate 10000 random numbers between 0 and 1 what do you call that distribution?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#24 2013-03-07 23:40:37

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,606

Re: Simulations

Uniform.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#25 2013-03-07 23:41:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,882

Re: Simulations

You will need to know how to get a histogram of those 10000 points with 10 boxes. Do you know how?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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