How'd you get variance? Or is this the conditional variance of not getting a one?
Ok I redid my math at the beginning all over agian. Some critical mistakes I made early on.
hhm... This makes some dangerous sense to me. Revisions again!!
I took the easy way out and computed it all in a spreadsheet.
I may have still have gotten it wrong, so I'll try again like everybody else:
Variance = E(X**2) - E(X)**2
E(X) = (A + B * Y) / 10
A = sum(k=2..9, k) = 44
B = sum(k=2..10, k) = 54
Y = sum(k=1..inf, (2/10)**k) # doubled roll, one-tenth chance
= 0.2 * sum(k=0..inf, (2/10)**k)
= 0.2 / (1 - 0.2) # geometric series
= 0.25
E(X) = (44 + 54 * 0.25) / 10
= 4.4 + 1.35
= 5.75
So far, so good.
E(X**2) = (C + D * Z) / 10
C = sum(k=2..9, k2) = 284
D = sum(k=2..10, k2) = 384
Z = sum(k=1..inf, (4/10)**k) # quadrupled roll (2 squared), still one-tenth chance
= 0.4 * sum(k=0..inf, (4/10)**k)
= 0.4 / (1 - 0.4) # geometric series
= 2/3
E(X**2) = (284 + 384 * 2/3) / 10
= 28.4 + 25.6
= 54
OK.
Variance = 54 - 5.75**2
= 54 - 33.0625
= 20.9375
Standard deviation ~= 4.576
YAY! for yet another different variance.
Ok, so your Variance is pretty much the same as what I have, (roundoff error on my part probably.)
Seems like a pretty simple way to learn with a potentially high pay off.
As for the cost of constantly studying from vis, yeah, it's probably cheaper to get tracti.
I, the guy without the math skills, knew that already 
Despite the great potential, most of my characters will exhaust all summae and then all tractii before studying from vis. Vis is not so abundant that I would use it so easily rather than saving it for a better project. There is still greater chance of a botch than a roll over 40 (1% chance of botch, 0.158% chance of a 40 or higher).
thus my answer to the poll, despite the potential for huge rewards, the risk and cost are high enough that I won't do it until I need to.
Fair enough. My first thought was off.. It does not seem worth it until as a last resort!
I'm thinking about using the imaginative learner virtue (if they still have it, existed in 4th) being used for rerolling on 1s AND 2s (as a house rule of course) for a much bigger chance of high payoffs.
I've got to admit that in the absence of a good suma my character is intending on studying Perdo from vis as his present score is 4 + 2xp. That means that even if he rolls a zero he only gets one botch die and the cost of one pawn is light for the covenant (because we have a perdo source).
Ah, but not "last" in the sense of time. Best to do it the way I suggested and Erik is planning. You get more per pawn at low Art scores, less at high ones. Tractatus can be used whenever. Use vis in the absence of good summas in the middle region and perhaps a little higher if you're lacking many tractatus. If you're short on tractatus, save them for later.
Also, note that people under-calculated the chance for a botch. I've seen suggestions of 1%. But that's the minimum chance. The maximum chance nears 10% but is unlikely. But let's say you have an Art score of 26. You need 6 pawns of vis and 6 botch dice. That gives you about a 4.69% chance of a botch. Your chance of at least a double-botch is about 1.14%. And remember that you'll see the higher Art scores later in your career. That raises the chance of entering twilight and raises the chance that the twilight will last a significant amount of time. By the time you're at a warping score of 6 a double-botch will probably cost you a season about half the time. (Remember that you'll want to be in a reasonably high aura to help the study quality.) So as the costs get worse your average also decreases slightly and your time commitment to get the experience rises on average, which effectively means the average drops further.
So use vis second (after summas, before tractatus) as a last resort (in an Art that you'll push beyond what all available tractatus can handle).
Chris
I agree with Fruny's calculations, I remember doing this math too a while back. I won't go into WHY I was doing this, lest I should sound like a probability textbook or some condescending math buff. But while I'm on the topic, I am wondering why anyone would be interested in the VARIANCE of this distribution, unless you're just interested in the mental exercise (which, I admit, can be fun, if you are the right type of person). The average, sure, that's good to know, and it's probably also good to know the pdf and the median. But the probability density function for the stress die is nowhere near Normal, or t-distribution, or even uniform (although that is closest), so you'll never be able to make the kind of assertions about confidence intervals that you would with those well-known distributions. In short, the point is, $%#& variance.
As far as studying from vis versus tractati (of course summas should be read first), I would also go with vis when the art is at a low level, and then turn to the tractatus after the level is above 10. It only costs 1 or 2 vis per season to study when your arts are low, but if you wait until after you've read every possible book and have a 26 and want to learn more from vis, you're going to have to bust a lot more than that, which is expensive and dangerous.
The variance gives the standard deviation, which gives you an idea of the spread, that's all I was interested in... That and the mental excercise.
For example, if we allow rerolls on 1s and 2s then a double, we get a expected value of 7.67 and a st dev of 10. Pretty significant difference. Given the otherwise slow pace of the game, I'd be for allowing rerolls on 1's and 2's, as this makes up for the chance of botching, and the opportunity cost of using vis that could have been used for something else.
Sorry if I hurt your feelings. I just remember really hating my first semester of probability. Now I'm okay with it I guess. But I know that standard deviation means a typical deviation from the mean, and when people say it is, what, 4 point something, that's telling us the stress die will typically give us a number from about 1 to 10, which I consider kind of obvious in this case, as it's a 10-sided die. And moving out to multiple standard deviations from the mean is not helpful. You won't roll any negative numbers, and the chance of getting either 10, 12 or 14 is actually worse than your probability of getting 16, 18, or 20.
Chameleon, no you didn't hurt my feelings no worries.
The median is pretty easy to eyeball here, unmodified, the median is 5 or 6 I believe.
I think the pdf would be neat to calculate, but I don't think it would be possible to give in any kind of closed form.
This analysis is correct if you assume perfect knowledge of the amount of books you'll ever need, and of the availability of books when you need them.
Otherwise, it might still be worth studying Ignem from books immediately, because you will benefit more right now - and who knows, you might never run out of books in the future, whether because you die of a heroic death, or because you create the greatest Academy on Ignem in the history of the Order (which will keep churning out books faster than you can read them), or because you might simply find that Herbam is more interesting than Ignem.
Then, there is the possibility that you will indeed run out of books. At that point, it might still be worth studying from vis, though inhindsight* it would have been better to study from vis first and from books then. But this does not invalidate the fact that without foreknowledge, the most rational course of action might be to study immediately from books.
That said, I would add that in all sagas (post 3rd edition) I've played in, studying from vis is extremely rare. With an Order of the current size, there are more good quality tractatus around than you'll ever need unless you want to blow your Herbam score past 70 (and why would you ever want to do that?). That's less than 2500 xp, probably less than 250 tractatus on Herbam. It might seem like a lot, but remember there's over a thousand magi in the Order, there were probably at least as many who are now dead, and every magus who trains an apprentice really wants a score of 5+ in every art - and that making two perfect copies of a book, even to the point of endowing them with a +1 bonus from resonant materials, costs less than 1 pawn of vis to most covenants.
Yes, this assumes a relatively organized Order, but in my opinion, if it can get to the level of organization that allows for Redcaps and Quaesitors and Tribunals ... inter-tribunal book trade should be the norm. Now, sagas set in the early days of the Order are a different thing altogether, as are ones set in fantastic, isolated locations, or ones centering on groups of young ne'er-do-well magi whom nobody would want to see with a high score in Ignem for all the vis in the world.
Learning from vis is senseless.
The 3rd edition was better there were no tractatus so vis learning was a must after magi learned the summas. It was cheaper and effective.
It is for high levels. For low levels it can be quite OK.
Cheers,
Xavi