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Math Help - Skewness

  1. #1
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    Question 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
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  2. #2
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    Quote Originally Posted by okthenso View Post
    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 E(X^3). Use the definition.

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


    But on the bright side, note that:

    E(X) = \mu = p

    E(X^2) = \sigma^2 + [E(X)]^2 = \sigma^2 + \mu^2 = p(1-p) + p = 2p - p^2.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Yes, you have to calculate E(X^3). Use the definition.

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


    But on the bright side, note that:

    E(X) = \mu = p

    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]?
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    Quote Originally Posted by okthenso View Post
    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.
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  5. #5
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    Quote Originally Posted by okthenso View Post
    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 E((X - p)^3) = (0-p)^3 q + (1 - p)^3 p = p q (q - p) ....

    Similarly for kurtosis.
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