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Author Topic: Random Stuff Part III  (Read 650390 times)

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silverspawn

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Re: Random Stuff Part III
« Reply #2150 on: December 20, 2016, 08:46:03 pm »
+1

no, that's perfectly normal :P

pacovf

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Re: Random Stuff Part III
« Reply #2151 on: December 20, 2016, 08:52:13 pm »
+2

Define "equal" and "as good as" first.
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Witherweaver

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Re: Random Stuff Part III
« Reply #2152 on: December 20, 2016, 10:23:42 pm »
+2

Wait...

D10-1 is equal to D8

D6+1 is equal to D8

Is a roll of D10-1 as good as a roll of D6+1?

We taught you how to fish.  We gave you the tools.  Now go catch your fish.
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eHalcyon

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Re: Random Stuff Part III
« Reply #2153 on: December 20, 2016, 10:30:20 pm »
+6

Wait...

D10-1 is equal to D8

D6+1 is equal to D8

Is a roll of D10-1 as good as a roll of D6+1?

We taught you how to fish.  We gave you the tools.  Now go catch your fish.

Go fishing:
Rolled 1d8 : 1, total 1


You feel a nibble and begin reeling.  The specimen is powerful and puts up a fight.  It drags you deep into the depths.  As the water fills your lungs, you see that the beast that defeated you was... a lifeless old boot.
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Kirian

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Re: Random Stuff Part III
« Reply #2154 on: December 21, 2016, 12:38:59 pm »
+5

no, that's perfectly normal :P

Only if you roll a ton of dice together all at the same time, then consider the limit as the number of dice goes to infinity.
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Witherweaver

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Re: Random Stuff Part III
« Reply #2155 on: December 21, 2016, 12:44:37 pm »
+3

Is there a misconception here that having equal means is a sensible thing to consider as 'equality' of two random variables?  Because that's quite bad.  Simply take X to be a uniform distribution on a discrete space of, say, 1e100 elements and Y to be the uniform distribution on the space {E(X)}.  X and Y have the same mean, but will almost never agree.
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Witherweaver

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Re: Random Stuff Part III
« Reply #2156 on: December 21, 2016, 12:47:44 pm »
0

If you can match means and variances (that is, matching first moment and second moment), you're doing much better.  If you can match more moments, the distributions are closer.

Basically

E(X) = E(Y)

is weak

E(X^n) = E(Y^n) for all n= 1, 2, ..., N

for a decent size N is much better. 
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Kuildeous

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Re: Random Stuff Part III
« Reply #2157 on: December 21, 2016, 11:07:36 pm »
+2

Yes, cones of shame come in tiny sizes too. She ripped open her incision, and now she has to pay the price.



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liopoil

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Re: Random Stuff Part III
« Reply #2158 on: December 22, 2016, 12:03:57 am »
+1

If you can match means and variances (that is, matching first moment and second moment), you're doing much better.  If you can match more moments, the distributions are closer.

Basically

E(X) = E(Y)

is weak

E(X^n) = E(Y^n) for all n= 1, 2, ..., N

for a decent size N is much better. 
So, I've never taken a real statistics class, only "AP" Statistics. Even there we still talked about how two distributions can have different spreads, i.e. variance. I suppose this is the second moment, and I looked up how it's defined: Var(x) = E((E(x) - x)^2), which makes sense certainly. This doesn't look like what you wrote here, but I like the idea that you're just talking about how the expectation of the variable to a power is different from the expectation to a power. So then I thought, based on your post, before I saw the above definition, that maybe Var(x) = E(x^2) - E(x)^2, since that is always positive and seems to measure spread. But now I notice:

E((E(x) - x)^2) = E(E(x)^2 - 2x E(x) + x^2) = E(x)^2 - E(2x E(x)) + E(x^2) = E(x^2) - 2 E(x) E(x) + E(x)^2 = E(x^2) - E(x)^2

I think that's pretty neat. Can we define the n-th moment to be E(x^n) - E(x)^n? Wait, but that doesn't work for n = 1 at all. So then how would we define it analogously to the E((E(x) - x)^2) definition of variance? What even is the third moment? Maybe it's something like E(Var(x) - (E(x) - x)^2)), but wait that's zero, so on a whim, maybe we square the inner bit? Then it's:

E((Var(x) - (E(x) - x)^2))^2) = E(Var(x)^2 - 2 Var(x) (E(x) - x)^2 + (E(x) - x)^4) = E((E(x) - x)^4) - Var(x)^2 = E((E(x)^2 - 2x E(x) + x^2)^2) - Var(x)^2

Hmmm, that doesn't really work. Okay, maybe it's E((E(x) - x)^3) = E(E(x)^3 - 3x E(x)^2 + 3x^2 E(x) - x^3), which is E(x)^3 - 3 E(x)^2 E(x) + 3 E(x) E(x^2) - E(x^3) = E(x)^3 - E(x^3) + 3 E(x) Var(x)

I like this because if X and Y have the same mean and variance then the only different term is E(x^n). So is the n-th moment given my E((E(x) - x)^n)? I still don't have an intuitive feel for what it should actually say about the distribution in general, and also this does not work for n = 1.

EDIT: I looked it up, and looks like the n-th moment is indeed E((E(x) - x)^n), neat. Just ignore the first moment I guess. I also learned just now that the third moment measures skewness, which makes sense actually. If E(x) = 0, then the direction of skew is determined by the sign of the third moment I would think. Well, that was a fun distraction
« Last Edit: December 22, 2016, 12:15:53 am by liopoil »
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Witherweaver

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Re: Random Stuff Part III
« Reply #2159 on: December 22, 2016, 12:22:16 am »
+2

Yes,

Var(X) = E(X^2) - E(X)^2,

which is why I said matching the first two moments is matching mean and var.  (Well, I implied it, I guess.)  Var is 'basically' the same as the second moment, because once you have the second moment, you have the var.  (You've firstly calculated the first moment, of course.)

The nth moment is, by definition, E(X^n).  Well, this is really the 'raw' moment, but generally we say moment to mean raw moment.  But look at var.  It's just shifting the second moment by the mean (squared, so dimensionality works).  SO it's really a measure of the second moment, two, just normalized in some way.  In fact it's called a central moment, because it's the (second) moment about the mean.  (As you say, Var(x) = E((X-E(X))^2).

The third and fourth moments have some intuition.  The third (standardized) moment is called skewness (roughly, measure of how not symmetric it is), and the fourth (standardized) moment is called kurtosis.  This one is hard to understand, but it basically has to do with the tail distribution (how fat is it).  These are not the raw moments, but rather standardized, similar to how the var is (though these are not only shifted, but also normalized by the standard deviation in some way).  See the chart at https://en.wikipedia.org/wiki/Moment_(mathematics).

Beyond fourth, it's hard to get much qualitative description.   
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Witherweaver

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Re: Random Stuff Part III
« Reply #2160 on: December 22, 2016, 12:29:38 am »
+1

Oh, and also see moment generating functions: https://en.wikipedia.org/wiki/Moment-generating_function

If two random variables have the same moment generating function, they are the same (almost everywhere). 

If the moment generating function is analytical*, then it can be written as its Taylor series.  But the coefficients in the Taylor series are derivatives of the function (at 0), which is (by definition of the moment generating function), the corresponding moment of the variable.  (That's why it's called 'moment generating function'.)  So, if two distribution agree at all moments, E(X^n) = E(Y^n) for all n in N, then they have the same moment generating function, so they are the same.  (As long as everything converges, which it should as long as these expectations exist, or at least in most regular cases.)

*This should actually be the case just by definition of the MGF as m_X(t) = E(e^{t*X}).

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pingpongsam

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Re: Random Stuff Part III
« Reply #2161 on: December 22, 2016, 09:26:12 am »
0

Sentry.

Sentry draws first, so doesn't combo with Pearl Diver's topdeck effect!

Unless you play 2 Pearl Divers!  ::)
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Polk5440

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Re: Random Stuff Part III
« Reply #2162 on: December 22, 2016, 11:21:35 am »
+2

I will interpret.

We've talked about this before, but ...
Dice Dice Math Dice
Dice Math Dice Dice
DIIIIIIICE Math Dice?
Is this how I use math to beat my friends at D&D with dice?

I don't know about D&D, but it is true that...
The expected roll of a D10 minus 1 is equal to the expected roll of a D8. (5.5-1=4.5)
The expected roll of a D6 plus 1 is equal to the expected roll of a D8. (3.5+1=4.5)
The expected roll of a D6 plus 1 is equal to the expected roll of a D10 minus 1. (3.5+1=5.5-1)
But you can't really say that a roll of a D6 plus 1 is "as good as" a roll of D10 minus 1. What "as good as" means depends on the situation!

Deja vu. You should be able to calculate things like this yourself now.

Math Fact: Having equal means is not enough for two random variables to have identical distributions.

Look, I know lots of math.

Rats!

I don't know that much math, but I am learning.

Wow! Look at all this math I know! Way more than you ever need to know.

I jest.  ;)
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silverspawn

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Re: Random Stuff Part III
« Reply #2163 on: December 22, 2016, 11:25:48 am »
0

Okay, how do you have such a large avatar? I need to know.

Polk5440

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Re: Random Stuff Part III
« Reply #2164 on: December 22, 2016, 12:16:34 pm »
0

Okay, how do you have such a large avatar? I need to know.

It's like I'm cursed with a superpower I don't want. I would prefer the forum to be text only -- stripped of all memes, avatars, and annoying signatures. I turn off as much as I can in preferences. Yet I am the person who gets the largest avatar of them all.

(Actually, I think it's because I selected an Isotropic image from the menu in preferences.)
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Awaclus

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Re: Random Stuff Part III
« Reply #2165 on: December 22, 2016, 12:17:19 pm »
0

Yeah, the Iso art is larger than anything else you can possibly have as an avatar.
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silverspawn

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Re: Random Stuff Part III
« Reply #2166 on: December 22, 2016, 12:17:38 pm »
0

That kind of begs the question of why you have an avatar at all.

Awaclus

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Re: Random Stuff Part III
« Reply #2167 on: December 22, 2016, 12:18:16 pm »
+2

That kind of begs the question of why you have an avatar at all.

http://begthequestion.info/
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silverspawn

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Re: Random Stuff Part III
« Reply #2168 on: December 22, 2016, 12:28:05 pm »
0

That kind of begs the question of why you have an avatar at all.

http://begthequestion.info/

http://lesswrong.com/lw/nq/feel_the_meaning/

This begs the question

No it doesn't.

http://begthequestion.info/

yes it does, because that's how people use it, and it means what people think it means. It doesn't matter if it originally was meant to be something else

protesting an invevitable trend in words does not make language more exact. It's just pointless.

http://scientiststhesis.tumblr.com/post/41711103568/the-hidden-meaning-of-words

Watno

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Re: Random Stuff Part III
« Reply #2169 on: December 22, 2016, 12:33:23 pm »
0

That kind of begs the question of why you have an avatar at all.

http://begthequestion.info/

Why would the people on this site get to decide what begging the question means?
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Polk5440

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Re: Random Stuff Part III
« Reply #2170 on: December 22, 2016, 01:10:32 pm »
0

That kind of begs the question of why you have an avatar at all.

I was exaggerating to make a joke. I have have avatars and sigs turned off, but don't mind the use of avatars. It was a few people who had offensive images and really annoying sigs a while ago that caused me to turn them off initially and I've never turned them back on.
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Witherweaver

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Re: Random Stuff Part III
« Reply #2171 on: December 22, 2016, 01:26:43 pm »
+3

That kind of begs the question of why you have an avatar at all.

http://begthequestion.info/

Why would the people on this site get to decide what begging the question means?

It really begs the question, doesn't it?
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Witherweaver

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Re: Random Stuff Part III
« Reply #2172 on: December 22, 2016, 01:28:15 pm »
+1

...

In Liopoil's case, it is not way more than he'll ever need to know; it's precisely a subset of what he will need to know :P
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silverspawn

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Re: Random Stuff Part III
« Reply #2173 on: December 22, 2016, 01:28:47 pm »
0

That kind of begs the question of why you have an avatar at all.

http://begthequestion.info/

Why would the people on this site get to decide what begging the question means?

It really begs the question, doesn't it?

Yes it does!

Kirian

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Re: Random Stuff Part III
« Reply #2174 on: December 22, 2016, 02:46:25 pm »
0

That kind of begs the question of why you have an avatar at all.

http://begthequestion.info/

Why would the people on this site get to decide what begging the question means?

They don't decide, they're merely reminding you of the stance of prescriptivist linguists, and explaining why they side with them rather than with descriptivists on this point.
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