>>>>> As a closed-form expression.
>>
>> Right. So what you're actually saying is that you can't compute the 
>> confidence interval exactly using a finite number of applications of a 
>> particular arbitrarily chosen set of functions - the elementary 
>> functions.
> 
> That is indeed the definition.
> 
> This obviously doesn't matter for scott's case since he just wants a 
> numeric answer and Excel seems to have erf built in, I just wanted to 
> point out that far from being "relatively easy" to work out, the 
> cumulative distribution function of a Gaussian is generally the first 
> function people encounter which is impossible to actually compute in 
> closed form (normally this will come up in first or second year calculus 
> at some point).  I'll agree that at first glance it certainly *looks* 
> like it would be simple to compute though, which is part of why it's 
> interesting.

Well, the sine and cosine functions can't be computed by a finite number 
of additions, subtractions, multiplications, divisions and exponents. 
But since sine and cosine are included in the arbitrary "set of 
permissible functions", they don't count. But erf is not, so it does count.

To me, it seems that approximating erf (or its inverse) is no harder 
than approximating a sine or cosine - and we do that all day.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 <voi### [at] devnull> wrote:
> Well, the sine and cosine functions can't be computed by a finite number 
> of additions, subtractions, multiplications, divisions and exponents. 

  They can if you allow the parameters to be complex numbers (which is why
trigonometric functions are included in the set of elementary functions).

-- 
                                                          - Warp

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Warp wrote:
> Orchid XP v8 <voi### [at] devnull> wrote:
>> Well, the sine and cosine functions can't be computed by a finite number 
>> of additions, subtractions, multiplications, divisions and exponents. 
> 
>   They can if you allow the parameters to be complex numbers (which is why
> trigonometric functions are included in the set of elementary functions).

OK, but that just brings us back to the fact that the exponential 
operator has to be approximated. ;-)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Invisible wrote:
> http://office.microsoft.com/en-us/excel/HP052090051033.aspx

Nice, that link crashed my browser :) Clearly Microsoft and KDE don't like 
each other.

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Orchid XP v8 wrote:
> Warp wrote:
>> Orchid XP v8 <voi### [at] devnull> wrote:
>>> Well, the sine and cosine functions can't be computed by a finite 
>>> number of additions, subtractions, multiplications, divisions and 
>>> exponents. 
>>
>>   They can if you allow the parameters to be complex numbers (which is 
>> why
>> trigonometric functions are included in the set of elementary functions).
> 
> OK, but that just brings us back to the fact that the exponential 
> operator has to be approximated. ;-)
> 

Did you read the Wikipedia page on closed form expressions?  It's not 
very long and all of these points are addressed there.

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Nicolas Alvarez wrote:
> Invisible wrote:
>> http://office.microsoft.com/en-us/excel/HP052090051033.aspx
> 
> Nice, that link crashed my browser :) Clearly Microsoft and KDE don't like 
> each other.

Crashed it? Or just upset it?

Microsoft may use proprietry features and write terrible markup, but if 
a browser crashes when fed the wrong data, methinks the browser should 
be rewritten to fail gracefully. ;-)

The only unusual thing I can see about the page [other than too much 
JavaScript] is a flash advert.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Kevin Wampler wrote:

> Did you read the Wikipedia page on closed form expressions?  It's not 
> very long and all of these points are addressed there.

...OK, just read it. Doesn't seem to address very much.

Also, I've often wondered how the **** you compute something like the 
Gamma function or the Bessel-J function. I mean, have you *seen* the 
definition?!

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 wrote:
> Kevin Wampler wrote:
> 
>> Did you read the Wikipedia page on closed form expressions?  It's not 
>> very long and all of these points are addressed there.
> 
> ...OK, just read it. Doesn't seem to address very much.

I was thinking of:



trigonometric functions and inverse trigonometric functions)."

and

"For purposes of numeric computations, being in closed form is not in 
general necessary, as many limits and integrals can be efficiently 
computed."

Which seem to exactly cover both of the points you recently mentioned, 
but perhaps I misunderstood what points you were making.


> Also, I've often wondered how the **** you compute something like the 
> Gamma function or the Bessel-J function. I mean, have you *seen* the 
> definition?!

Yes.  If you take a look at the Wikipedia article for the gamma function 
you'll see that it includes a couple of nice representations in terms of 
infinite products.  The Bessel functions seem a bit more involved, but 
it has representations in terms of a sum over terms involving the gamma 
function, a hypergeometric series, and a recursive relation to a 
continued fraction -- so there seem to be many ways to go about 
computing it (I didn't look up what the standard approach in practice was).

I know that you say that you have trouble digesting the Wikipedia 
articles on these in full (entirely understandable, they are very dense 
and not always well written) but you still can answer these questions by 
spending a minute or two just skimming over them without trying to 
understand everything.

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>> ...OK, just read it. Doesn't seem to address very much.
> 
> I was thinking of:
> 


> trigonometric functions and inverse trigonometric functions)."
> 
> and
> 
> "For purposes of numeric computations, being in closed form is not in 
> general necessary, as many limits and integrals can be efficiently 
> computed."
> 
> Which seem to exactly cover both of the points you recently mentioned, 
> but perhaps I misunderstood what points you were making.

Oh, I see. I thought it was going to say something more insightful than 
that. (How the hell do you compute a limit or an integral anyway?)

>> Also, I've often wondered how the **** you compute something like the 
>> Gamma function or the Bessel-J function. I mean, have you *seen* the 
>> definition?!
> 
> Yes.  If you take a look at the Wikipedia article for the gamma function 
> you'll see that it includes a couple of nice representations in terms of 
> infinite products.

But you can't compute an infinite product.

> The Bessel functions seem a bit more involved, but 
> it has representations in terms of a sum over terms involving the gamma 
> function, a hypergeometric series, and a recursive relation to a 
> continued fraction -- so there seem to be many ways to go about 
> computing it (I didn't look up what the standard approach in practice was).

Uh...OK.

> I know that you say that you have trouble digesting the Wikipedia 
> articles on these in full (entirely understandable, they are very dense 
> and not always well written) but you still can answer these questions by 
> spending a minute or two just skimming over them without trying to 
> understand everything.

Heh. Some of them explain things quite clearly, but others are 
incomprehensible. That's the trouble with a reference source written by 
bored Internet surfers; first it's a reference, not an introduction, and 
second the quality is *highly* variable. ;-)

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>> Not really more complex, just a game that is played repeatedly and you 
>> have a certain % chance of winning (say 57%).  I just want to know if you 
>> play 10,100, 1000 times what is the *likely* number of wins in some kind 
>> of worst case and best case scenario.  ie it's obviously possible to win 
>> every single game, but that is *really* unlikely, I want to know how many 
>> you'd win with 90% or 99% confidence.
>
> Right. So it's ye olde "2SD = 95% confidence" rule then.

Indeed.  It turns out in my game that you break even if you win 54% of the 
time, and in practise you can only realistically expect to win 57% of the 
time if you are good at the game.  These two facts mean that after even 1000 
games there is still a huge variation in how much profit you can expect to 
make - 95% confidence you will make between -7 and 607 arbitrary units of 
profit.

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