Thread: Skewness

have to find the mean, variance and skewness and kurtosis for various distributions.
Take the bernoulli distribution where the mean (u) is p and the variance is p(1-p)

I'm having trouble with the skewness. I know the formula for it...

E[(x - u)^3] / sd^3

and that I cube out the top to get E[x^3 - 3x^2u etc...

My problem where to go from here. I'll end up with a E[x^3], do I have to go back and calculate that then, and the same for every other E[...] I'll end up with?

Also, is the bottom line just the variance to the power of 3/2?

Some similar help on the kurtosis would be useful is possible.
Thanks

Originally Posted by okthenso

have to find the mean, variance and skewness and kurtosis for various distributions.
Take the bernoulli distribution where the mean (u) is p and the variance is p(1-p)

I'm having trouble with the skewness. I know the formula for it...

E[(x - u)^3] / sd^3

and that I cube out the top to get E[x^3 - 3x^2u etc...

My problem where to go from here. I'll end up with a E[x^3], do I have to go back and calculate that then, Mr F says: Yes.

and the same for every other E[...] I'll end up with?

Also, is the bottom line just the variance to the power of 3/2? Mr F says: Yes.

Some similar help on the kurtosis would be useful is possible.
Thanks

Yes, you have to calculate $\displaystyle E(X^3)$. Use the definition.

And to get kurtosis you'll need to calculate $\displaystyle E(X^4)$ as well.


But on the bright side, note that:

$\displaystyle E(X) = \mu = p$

$\displaystyle E(X^2) = \sigma^2 + [E(X)]^2 = \sigma^2 + \mu^2 = p(1-p) + p = 2p - p^2$.

Originally Posted by mr fantastic

Yes, you have to calculate $\displaystyle E(X^3)$. Use the definition.

And to get kurtosis you'll need to calculate $\displaystyle E(X^4)$ as well.


But on the bright side, note that:

$\displaystyle E(X) = \mu = p$

$\displaystyle E(X^2) = \sigma^2 + [E(X)]^2 = \sigma^2 + \mu^2 = p(1-p) + p = 2p - p^2$.

Thanks... one last question.... when I have the E[x^3 - 3x^2p...]

Will the second part of that be 3pE[x^2], or do I calculate E[3x^2p]?

Originally Posted by okthenso

Thanks... one last question.... when I have the E[x^3 - 3x^2p...]

Will the second part of that be 3pE[x^2], or do I calculate E[3x^2p]?

E(aX + b) = a E(X) + b.

Originally Posted by okthenso

have to find the mean, variance and skewness and kurtosis for various distributions.
Take the bernoulli distribution where the mean (u) is p and the variance is p(1-p)

I'm having trouble with the skewness. I know the formula for it...

E[(x - u)^3] / sd^3

and that I cube out the top to get E[x^3 - 3x^2u etc...

My problem where to go from here. I'll end up with a E[x^3], do I have to go back and calculate that then, and the same for every other E[...] I'll end up with?

Also, is the bottom line just the variance to the power of 3/2?

Some similar help on the kurtosis would be useful is possible.
Thanks

Alternatively, you could use the pmf of the Bernoulli distribution to directly get $\displaystyle E((X - p)^3) = (0-p)^3 q + (1 - p)^3 p = p q (q - p)$ ....

Similarly for kurtosis.