Question

Given the first 4 central moments of a random variable x are 0,0.566,0.393 and 0.756 respectively, find the moment coefficient of skewness and kurtosis and interpret them.

Answer 1

Solution:

First central moment$=E(X)=\mu=0$

Second central moment$=E[(X-\mu)^2]=E[(X-0)^2]=E[(X)^2]=0.566$

Third central moment$=E[(X-\mu)^3]=E[(X-0)^3]=E[(X)^3]=0.393$

Fourth central moment$=E[(X-\mu)^4]=E[(X-0)^4]=E[(X)^4]=0.756$

Now, $Var[X]=\sigma^2=$ Second central moment$=0.566$

So, $\sigma=\sqrt{0.566}=0.752329$

Next, $Skewness(X)=\dfrac{E[(X-\mu)^3]}{\sigma^3}=\dfrac{0.393}{0.752329^3}=0.92293$

And, $Kurtosis(X)=\dfrac{E[(X-\mu)^4]}{\sigma^4}=\dfrac{0.756}{0.752329^4}=2.35988$

We know that for skewness is that if the number is greater than +1 or lower than –1, this is an indication of a substantially skewed distribution. Here, we get Skewness=0.92293 <+1, this shows there is no skewed distribution.

For kurtosis, we know that if the number is greater than +1, the distribution is too peaked.

Here, Kurtosis=2.35988>+1, this shows the distribution is too peaked.