IAI May 2010

The very first two questions to this paper ask about the empirical skewness coefficient or to find the probability empirically...
I want to know where in our stats pack have we been taught this?
How on earth is a student supposed to know what this thing means..!!

 

I don't have the paper and haven't seen the question, but empirically simply means 'by experiment'

Without any prior knowledge, if you flip a coin 100 times and it comes up heads 70 times, empirically you would say there is a 70% probability of heads in any given flip.

You use sample formula to find other empirical results

EG.
empirical mean = [sum x(i)]/n (first moment about origin for a sample)
empirical SD = sum (x (i) - u)^2 / n-1 (second moment about mean for a sample)
empirical skew = third moment about mean for a sample

 

http://actuariesindia.org/downloads/exampapers/May%202010/QP/CT3_IAI_QP_0510.pdf

Find the link

In the second question, n has been used to compute the sample variance instead of (n-1). Why??

 

You use n-1 for a sample; if you have the entire population, you use n. Have to agree I wouldn't consider this particularly obvious.

 

All will become clear by the end of the course! (or by the time you reach the chapter on estimation)

 

I have already completed the course and I know that we use n for population variance while n-1 is used for sample variance. But if you see the solution to the second question, n has been used to compute sample variance (which in turn has been used to compute empirical coefficient of skewness).

Find the link to the solution given in the institute's website.

http://www.actuariesindia.org/downloads/exampapers/May 2010/Sol/CT3_Solution_May 2010_Final.pdf

 

You are computing theoretical measures here. You are not working with samples. In fact if you notice in the solution there is no division by 'n' or 'n-1' involved here.

It may be worth revising Section 4.1 of the notes again.

 

This is all to do with the definition of skewness. Skewness is defined to use variance with an n instead of n-1. So here the variance with an n is calculated as part of the interim steps taken to calculate the skewness.

If the question had explicitly asked for the variance too, the answer would require you to use the n-1 version.

Technically both are consistent, they both converge to the population variance as n tends to infinity, it's just the n-1 version is unbiased, so more preferable.

 

Ya, got it.. Sorry, my mistake.
Thanks, bapan & nick.