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Math Help - Simplifying product of to sum of in likelihood equation

  1. #1
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    Angry Simplifying product of to sum of in likelihood equation

    Hi, I'm having problems with the veru bsics so I cant get any further on my questions. When given a function f(xlt). to find the likelihood you have to do the product of it. I dont understand how the product of sign changes to the summation sign.

    For example L(t)= product of( t^x)xe^(-tx)

    How do i simplify this to the sum of form?

    Im sorry about how its written I dont know how to put in the symbols.
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  2. #2
    MHF Contributor matheagle's Avatar
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    1) It's a product of f(x_i) where i goes from 1 to n
    IT's just the joint distribution function of n i.i.d. copies from that distribution

    2) My guess is that your asking about the LOG likelihood function
    It is what it's called, it is the logarithm (any base is fine) of the likelihood function
    LOGs turn products into sums.
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  3. #3
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    Using neyman pearson lemma

    The question was to use the neyman-pearson lemma for t_0>t_1

    So what I've done so far..No idea how wrong it is..

    L(t)= sumof(x_i)*t^(sumof(x_i))*exp(-sumof(tx)) but we can get rid of terms without t.

    So L(t_0)/L(t_1)=
    (t_0)^sumof(x_i)*exp(-t_0(sumof(x_i)))/
    (t_1)^sumof(x_i)*exp(-t_1(sumof(x_i)))

    =(t_0)^sumof(x_i)/(t_1)^sumof(x_i) * exp(-t_0(sumof(x_i))+t_1(sumof(x_i)))

    and this is <= K

    Since t_0>t_1, t_1-t_0<1 and t_o/t_1 <1

    so the most powerful test has critical region of the form sumof(x_i)>=K

    Does this look right?

    Once again sorry for the mess without the correct symbols.
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