CT6 Ch10 sec 1.3 exponential family for binomial

Discussion in 'CT6' started by Ark raw, Aug 8, 2017.

  1. Ark raw

    Ark raw Member

    In this section, we prove that binomial distribution belongs to the exponential family but once I went through the proof it stated that Z~(n,μ) then stated that Z=Y/n. And then we went on to prove that y belongs to the exponential family, I wanna know how proving Y belongs to exponential family proves that Z belongs exponential family?

    And 2ndly can we use the proof shown in this video in the exam? ''

    And I have a 3rd question is, when b(θ)=-ln(-θ), then how b'(θ)=-1/θ? shouldn't it be b'(θ)=1/θ

    Thank you
     

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    Last edited by a moderator: Aug 8, 2017
    Ark raw, Aug 8, 2017
    #1
  2. John Lee

    John Lee ActEd Tutor Staff Member

    We're not trying to prove that Z is a member of the exponential family - we're trying to prove that Y is. However, we re-parameterise it so that we can separate the scale parameter n from the mean \(\mu\).

    The video proof says there is no scale parameter which isn't the Core Reading approach. So no it wouldn't be marked correct.

    Differentiating ln f(x) gives f '(x) / f(x). So you'll have three minus signs.
     
    John Lee, Aug 11, 2017
    #2
    Ark raw likes this.
  3. Ark raw

    Ark raw Member

    But how does proving Y belongs to exponential family prove that binomial distribution also belongs to the exponential family?
     
    Ark raw, Aug 11, 2017
    #3
  4. John Lee

    John Lee ActEd Tutor Staff Member

    Because Y is a binomial.

    It might be re-parameterised - but it's still a binomial
     
    John Lee, Aug 11, 2017
    #4
  5. Ark raw

    Ark raw Member

    So Z~bin(n,μ) and Y=Z/n so Y is a binomial random variable w/ mean (natural parameter)=μ and scale parameter=n, so can we say that Y~bin(n,p) where μ=np? and since Z~bin(n,μ) then is μ<=1?
     
    Ark raw, Aug 12, 2017
    #5
  6. John Lee

    John Lee ActEd Tutor Staff Member

    We've scaled the binomial so that the new mean does take values between 0 and 1 (so it's a probability) so your second statement is correct. We then use this to model probabilistic outcomes (eg the incidence of a claim).
     
    John Lee, Aug 16, 2017
    #6

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