Questions about normal, z, t, chi, and F distributions

[sleepybird]

Hi David,
Sorry I have a number of questions regarding T2.a.Quantitative.
1. t-distribution variance = k/(k-2) and k = degree of freedom = n-1. This means t variance = (n-1)/(n-3)?
2. Are k = n-1 for all distributions, including chi and F? This would make chi variance = 2 (n-1)?
3. What are the mean and variance of the F-distribution? You mentioned those for all other distributions except for F.
4. Normal: 2 parameters (mean and variance); z: no parameter required!; t: defined by single parameter, k (or df); what about Chi and F?
5. You said in your video that F-test tests whether 2 variances are from the same population. Can you elaborate on that. My understanding from readings is that F tests whether the 2 variances from 2 different populations are equal. For example, analyst can use F-test to test whether the standard deviation of one industry sector (e.g., utility, 1st population) is greater than another industry sector (e.g., bank, 2nd population) using hypothesis testing.
Sorry for the long questions and thanks for your prompt response.

 

[David Harper CFA FRM]

Hi sleepybird,

1. Yes, true in test of sample mean (unknown pop variance) because k = n - 1, in that case. But (eg), if regression with two independent variables, then df = n - 2, for test of regression coefficient (also a test of sample mean).
2. No, not always true. Variance of chi-squared distribution is 2*df, where mean is conveniently
3. I don't know, i need to look up: http://en.wikipedia.org/wiki/F-distribution
4. Chi^2 (like t) has one param, d.f.; F has two params (two d.f.)
5. Yes, I agree with "F tests whether the 2 variances from 2 different populations are equal." My characterization is meant to mean the same exact thing, it's from a previous author, some say it that way. Although, I agree your statement seems technically superior.

Thanks, David

 

[sleepybird]

Thank you for your quick response.

4. I see. Since F test involves 2 populations, then params = 2 df. Thanks.

 

[sleepybird]

David,
One more clarification. In your video, you described the t-distribution as "fatter tail and more peaked than the normal distribution (leptokurtic)."
But my understanding is that the t-distribution is less peaked than the normal distribution.
So I google searched and found the below link. Second to last page (summary section) paragraph 3 also says it's less peaked.
Leptokurtic = more peaked and fatter tail.
Platykurtic = less peaked and thinner tail.

I find the combination of less peaked and fatter tail rather odd. Am I missing something here?

http://davidakenny.net/statbook/chapter_11.pdf

 

[David Harper CFA FRM]

H sleepybird,

Thanks for the PDF, very interesting the assertion on p 178!

I was stuck on this for a long time, too, until I read Carol Alexander (http://www.amazon.com/Market-Analysis-Quantitative-Methods-Finance/dp/0470998008). Note that it's an optical illusion due to the fact that a student's t (with variance > 1.0) is superimposed on a standard normal (variance = 1): you can't see the graphic, but when the variances are apples-to-apples, the student t is taller! (this is the only place i have ever seen it explained):

We emphasize that the Student t distribution (I.3.52) does not have unit variance, it has variance given by (I.3.53). So Figure I.3.15 is misleading: it gives the impression that the t distribution has a peak that is too low compared with the normal density. However, in Figure I.3.16 we compare a Student t density function with six degrees of freedom with the normal density having the same variance, i.e. the normal with mean 0 and variance 1.5, and then the leptokurtic character of the Student t density is evident. -- Carol Alexander, MRA Vol I

Thanks, David

 

[sleepybird]

Thanks. So for the exam, we stick to leptocurtic.

 

[David Harper CFA FRM]

Yes, the exam follows the aged Linda Allen (and conventional wisdom) in generally assuming financial returns are heavy tailed (leptokurtic). Further, the implied peakedness does not really fascinate us as it's a property of the "body" and risk is generally concerned with the tail.