Properties of Univariate statistics.

[zyQuzA0e5esy2y]

Hi PF, i have several questions about univariate statistics that doesn't seem to be covered in my notes or online, i hope the question is not redundant on the forums, but i ran a search and saw nothing.

In univariate statistics, you can have a PMF which is a discrete random variable (RV) and a PDF which is a continuous RV.

"We can estimate these quantities given a random sample of observations on a random variable, specifically, a random sample of n independently sampled observations on the random variable X is a set of random variable, each of which has the same distribution as X. That is, letting Fx(x) denote the CDF of Xi."

we can say that random variables, are independent and identically distributed (IID), since each observation has the same distribution, E(X) and variance are the same thus COV(Xi,Xj) = 0"

What happens if it was a PMF or is it not possible?

A normal distribution of method of moments tell us:
1st mom = E(X)
2nd mom = Variance
3rd mom = skewness
4th mom = Kurtosis
Does the skewness tell us the direction which the curve is skewed and if the E(X) and variance is on the left side or right side of the curve?

What is kurtosis and what does it tell us?
in my notes i have that the kurtosis tells me that it is a function of the first 4 moments which tells me the E(X), variance, skewness and kurtosis, but doesn't exactly tell me about kurtosis. could i possibly get an explanation?

 

[statdad]

"What happens if it was a PMF or is it not possible?"
Same - if your sample is from a continuous distribution or discrete distribution (your only two distinctions) is immaterial: you can obtain the same information about moments, etc.

Roughly speaking (you can get more details by googling kurtosis)
Kurtosis gives one way to indicate how close the data's distribution is to a normal distribution

• Normal distributions have kurtosis equal to zero
• A distribution "flatter" than a normal has negative kurtosis
• A distribution more strongly peaked than a normal has positive kurtosis

 

[zyQuzA0e5esy2y]

Thank you very much, this actually helped. i tried googling kurtosis but didn't understand it as much.