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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Hi;

Okay, take your time, I will wait.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi bobbym,

I think these must be right, finally!

1)

where 'n_a' is any of the number of cards available of digit 'a' and 'h' is the number of cards greater than 'a'.

2)

3)

4)

For (9,9,7),

And the expected sum of the highest three, using the above equations:

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Hi;

H = 24, how would you phrase that? Cards less then a?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi,

'h' is the number of cards having face value > 'a'

For the example, a=7 and 'h' includes all 8's, 9's and 10's: h=4+4+16

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Very good! I am checking the last one now and you have an important result!

Could you please explain what f\goes into what for your formula 4).

The example is (7,7,9)

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Yeah, but took quite a while to correct a mistake in my program!

For (7,7,9),

a=7, b=9, h=24, n7=n9=4

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Hi;

I will run it through your fourth formula.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Okay.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Hi;

Did you see the top part of post #2030?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi,

Yes, the top part of my reply was refering to my program for calculating the expectation.

I saw that was what you actually wanted!

Also saw that it was already asked in "help me", good question!

Would you like me to post my code in "computer math" ?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

I saw that was what you actually wanted!

You read the future.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

You read the future.

Nope, math.se!

Yes, I was talking about that thread.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

That was me. Why the heck did you not post in there?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

I have forgotten my login credentials there, had registered once a long time ago.

Never mind, you post the answers there you now know.

We can easily extend the formula for any number of picks. But more cases if we need sum of 'n' highest cards, with n>3. I wonder if the cases can be eliminated altogether!

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

They do not get those answers. They belong to you and it is your choice whether or not to share them.

I have been working on using order statistics for the problem but have not been successful.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

I don't mind you sharing the answer there.

The 10's have a different probability, that's the difficulty we have here.

Otherwise, we might have had a better answer.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

I am hazy on this whole idea so it might be wrong but there is a distribution called the Empirical Distribution which handles that.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Okay, I'll check that.

Taking a break now, see you later.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Okay, and thanks for coming in.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Here's a formula for probability of the order statistic:

E.g. For the probability that 6th highest number is 7 among 8 selected, we have:

I don't know a formula for joint distribution yet.

The literature on these look stupid without any example.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

Hi gAr;

I have had much trouble with the order statistic of an empirical distribution like these. What literature are you looking at?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi,

I had not tried order statistics of discrete distributions before, the formulas in the program had Vandermonde's identity in it.

Literature, whatever I could find online.

I tried the formula in wikipedia, but couldn't understand what to make of f(x) and F(x).

I also looked at "A first course on order statistics", doesn't have a single numerical example!

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

I think I remember that book, I had to throw it away for that very reason.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

If we can get a formula for joint distribution, our job for the expected sum will be done!

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,257

That has been difficult for me. Mathematica has a command to do all these type problems but when I try to have the command work on an empirical distribution like this one it fails. This has been the major problem, I can not get exact answers and must rely on simulations which are not close enough.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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