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Thread: Raw moments of 3-parameter Weibull distribution

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    Raw moments of 3-parameter Weibull distribution




    Hi,
    I know that the formula for computing the raw moments of 2-parameter Weibull distribution is:
    Mu'n=b^n*Gamma(1+n/c), where b and c are scale and shape parameters, respectively.

    However, I couldn't find any exact formula for a 3-parameter Weibull distribution. Is there any simple formula for it???

    Thank you in advance

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    Re: Raw moments of 3-parameter Weibull distribution

    First I need to make sure that the 3rd parameter you mentioned is the location parameter.

    https://en.wikipedia.org/wiki/Weibul..._distributions

    If that's the case, then we have the following:

    Let X \sim \text{Weibull}(k, \lambda, \theta) and Y \sim \text{Weibull}(k, \lambda, 0) be two 3-parameters Weibull random variable.

    Then
    Y is a 2-parameter Weibull random variable; and

    X \stackrel {d} {=} Y + \theta

    As you know raw moments

    m_n \triangleq E[Y^n] = \lambda^n \Gamma\left(1 + \frac {n} {k}\right)

    Therefore,

    E[X^n] = E[(Y + \theta)^n]

    = E\left[\sum_{r=0}^n \binom {n} {r} Y^r \theta^{n-r}\right]

    = \sum_{r=0}^n \binom {n} {r} m_r \theta^{n-r}

    = \sum_{r=0}^n \binom {n} {r} \lambda^r 
\Gamma\left(1 + \frac {r} {k}\right)  \theta^{n-r}

    This is not simple or very nice, but at least this is exact and can be computed.

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    Re: Raw moments of 3-parameter Weibull distribution


    Thank you so much for your reply.
    This is exactly what I needed, although not very simple (as you mentioned)

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