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Beta distribution
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====Higher moments==== Using the [[moment generating function]], the ''k''-th [[raw moment]] is given by<ref name=JKB/> the factor :<math>\prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} </math> multiplying the (exponential series) term <math>\left(\frac{t^k}{k!}\right)</math> in the series of the [[moment generating function]] :<math>\operatorname{E}[X^k]= \frac{\alpha^{(k)}}{(\alpha + \beta)^{(k)}} = \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r}</math> where (''x'')<sup>(''k'')</sup> is a [[Pochhammer symbol]] representing rising factorial. It can also be written in a recursive form as :<math>\operatorname{E}[X^k] = \frac{\alpha + k - 1}{\alpha + \beta + k - 1}\operatorname{E}[X^{k - 1}].</math> Since the moment generating function <math>M_X(\alpha; \beta; \cdot)</math> has a positive radius of convergence,{{cn|reason=proof that the radius of convergence is positive, not in Billingsley Section 30?|date=December 2024}} the beta distribution is [[Moment problem|determined by its moments]].<ref>{{cite book|last1=Billingsley|first1=Patrick|title=Probability and measure|date=1995|publisher=Wiley-Interscience|isbn=978-0-471-00710-4|edition=3rd|chapter=Section 30: The Method of Moments}}</ref>
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