Would anyone please elaborate the concept how we could use \(Skew(X)=\Sigma([{x_i}-{\mu}]^3)\) is equal to \(Skew(X)=E([{X_i}-{\mu}]^3)\)?
Skew(X)=E([Xi−μ]^3) is the population skewness. This is essentially the 3rd central moment (mu3) You can estimate the skewness from a sample using Σ([xi−x_bar]^3)/n. Not sure if Σ([xi−μ]^3) has any real significance !
The population skewness is E[(X-mu)^3], where X is the random variable and mu is its mean. By cubing the deviations from the mean, we retain the sign, so a positive skewness tells us that we have more values of X above the mean than below the mean, ie it is peaked to the left and has a long tail on the upper side. A negative skewness tells us that we have more values below the mean than above it, so it is peaked to the right with a long tail on the lower side.
Doubt raised because the way Part 1's Question No.1.2 and Question No.1.5(iii) have been solved. They might have eliminated the \(1/n\) in Question No.1.2 as it has no use there.
For Q1.2, you are required to compute sample skewness. Read the question it gives you information about a data set. For Q1.5, you are dealing things at the population level. You define skewness as expectation of some random variable dependent on X. Then it goes on to compute the expectation using first principles. I am not sure when you wrote your original post why you decided to drop the factor P(X = x) within the summation formula ! In summary, I do not see any issue here. Is this fine?
My question was completely different although. And I have not dropped the \(P(X=x)\) factor. There was no significance of \(1/n\) in Q.1.2 and that's why the Profession has eliminated that factor. I got little confused for that only. Now there is no issue at all. By the way, both the questions asked us to find the Skewness as far as I've understood. It's okay now. Thank you.
Yes, you are correct. 1/n has no significance in answering Q1.2 as we need to understand the sign of the answer and not the actual answer. If it asked you to compute skewness, you have to use (1/n). Yes, both questions will say compute skewness. You have to understand from the context if this is a population or a sample measure.
