Even as something as “simple” a t-distribution (or one with too many degrees of freedom) would fit ITA’s post about under representing tail risk-- if you use a normal distribution, the tails are always less fat than if you had used some kind of t-distribution, for example (i.e. looks a lot like a normal distribution but technically isn’t since you need more than mean and variance to accurately describe the distribution, you also need the degrees of freedom for a t-distribution to be accurately described).
Even then, it’s about proper application right?
This isn’t true, at least for t-distributions with degrees of freedom greater than 2. If the number of degrees of freedom is ν > 2, then the standard deviation is:
\sigma = \sqrt{\frac{\nu}{\nu - 2}}
Solving for dof gives:
\nu = \frac{2\sigma^2}{\sigma^2 - 1}
Therefore, the mean (= 0) and standard deviation uniquely determine a t-distribution when dof > 2. You don’t need to specify the dof in addition to the standard deviation.
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Well, it is true, and there are very important results regarding the t distribution with a single df! (Cauchy…cool stuff at 2, 3, 4 also…)
I’m not denying the relationship between df and sd under specific conditions, but what I said is generally true because you need to know the df to understand if the variance formula holds and what other important results you get at various other df. The normal distribution doesn’t have this result.
You did add specificity to what I said but I don’t think that makes what I said untrue, just not as general, no? (I think yours also ignores skew and kurtosis that have a relation to df?)
I think that my point is more general, however.
Many distributions can be characterized fully by giving their mean and standard deviation . . . as long as you know what type of distribution it is. A normal distribution is characterized fully by its mean and standard deviation . . . as long as you specify that it’s normal. A (continuous) uniform distribution is characterized fully by its mean and standard deviation . . . as long as you specify that it’s uniform. A Student’s t-distribution (with more than 2 degrees of freedom or more) is characterized fully by its mean and standard deviation . . . as long as you specify that it’s a Student’s t-distribution. And so on.
Some distributions, of course, are not characterized fully by only the mean and standard deviation . . . even if you know the specific family into which the distribution falls.
The only reason I harp on this is that I’ve seen far too many people say that a normal distribution can be characterized uniquely by its mean and standard deviation, suggesting that this is something unique to normal distributions, which isn’t remotely the case.
I think that my point is more general, however.
I trust your mathematical skills more than mine-- to me, mine sounded more general, and now I also see what issue you were concerned with below.
Many distributions can be characterized fully by giving their mean and standard deviation . . . as long as you know what type of distribution it is. A normal distribution is characterized fully by its mean and standard deviation . . . as long as you specify that it’s normal. A (continuous) uniform distribution is characterized fully by its mean and standard deviation . . . as long as you specify that it’s uniform. A Student’s t -distribution (with more than 2 degrees of freedom or more) is characterized fully by its mean and standard deviation . . . as long as you specify that it’s a Student’s t -distribution. And so on.
See your concern for sure now-- I was speaking in context which was about normal distributions and then my example of t distribution (but I still think needs to know the df to adequately describe how fat the tails are, for example).
Some distributions, of course, are not characterized fully by only the mean and standard deviation . . . even if you know the specific family into which the distribution falls.
The only reason I harp on this is that I’ve seen far too many people say that a normal distribution can be characterized uniquely by its mean and standard deviation, suggesting that this is something unique to normal distributions, which isn’t remotely the case.
Agreed, and my poor use of the word “since” does read as “this is the reason, otherwise it would be normal” which isn’t true because there are many other things that make it non normal (but again, I think it’s important to know the df (or that relationship) to see the proper t-distribution. I.e. you can’t do much with the information of mean and sd for the t-distribution without then transforming the sd to get the df.
I agree.
However:
- Whether the information is presented efficiently or not is a far different question than whether it is sufficient or not. If I tell you that I have a piece of wood whose (nominal) cross sectional dimensions sum to 6 inches with a difference of 2 inches, I have completely characterized a 2×4, although not as straightforwardly as I might have.
- The fact that most statistics students don’t know how to get a t-distribution’s standard deviation from its df or vice-versa suggests to me nothing more nor less than a shortcoming in the way that those students have been taught t-distributions. I, for one, don’t recall having been taught the standard deviation of a t-distribution; I wonder why that is.
Not, sure, either. I don’t know of many problems in real life where you have the degrees of freedom but not the standard deviation for most applications where you would employ the t-distribution. Maybe that has something to do with it?
Good convo, I almost forgot there used to be a time when conversations like this were commonplace on here.
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Just wait until Chad flips the switch to only allowing certain people in. It’ll either be great or zero activity
It could be both.
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is hotness normally distributed? some people think its uniform and that pisses me off.
Uniforms generally add about 1 point.
Buffett Just sold DAL btw. ah at 20
My entry was around $20. I almost sold when it hit the $30’s but decided to just go for the ride. We’ll see I guess. Year end one of us can be crowned the winner!
Markets on fire here. Always the case I wish I had bought more, but still making out OK unless we get a complete retraction. Happy with some of the positions I picked up for a 1 - 2 year hold.
Nerdy: Hope you followed your own advice early and loaded up on GOOG.
CEO: If you actually took all of your earlier trading profits and plowed them into SSO you’re a god damn hero (see what I did there?). I’d hire you to manage my money any day. Of course, the hurdle rate would be 60% annually, but I’m sure you could handle that.
Nice!
NERDY, DAL off to the races, BS > WB
Sorry guys, just getting all my victory laps in before the market collapses, lol
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All 3 of your initial ideas (DAL / USCR / DRI) are up significantly since that post. I think you wrote that at, or close to the low for each. I actually picked up a little USCR… been a good run.