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How to distribute product sign across base and exponent terms

  1. Jun 26, 2009 #1
    I am calculating the likelihood of a hierarchical model and am having trouble distributing the product sign.

    Here are the two expressions that I'm interested in:

    Product from j=1 to j_k of: (t_jk)^(sum(Z_ijk) + a_k - 1)


    Product from k=1 to K of: (b_k)^(j_k*a_k).

    The tricky part for me is distributing the product sign across the base and the exponent terms simultaneously. Is it possible?

  2. jcsd
  3. Jun 26, 2009 #2
    \prod_{j=1}^{J_k} t_{jk}^{(\sum_{l=1}^L Z_{ljk}) + a_k-1} =
    \left(\prod_{j=1}^{J_k} \prod_{l=1}^L t_{jk}^{Z_{ljk}}\right)
    \left(\prod_{j=1}^{J_k} t_{jk}\right)^{a_k-1}

    \prod_{k=1}^K b_k^{j_k a_k}
    \;[/tex] probably not much to do with this
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