Hey there! Sign in to join this conversationNew here? Join for free

Combined likelihoods for random samples Watch

Announcements
    • Thread Starter
    Offline

    0
    ReputationRep:
    How would I go about doing this question? Name:  ImageUploadedByStudent Room1453070079.520693.jpg
Views: 43
Size:  191.2 KB
    Offline

    13
    ReputationRep:
    (Original post by Bruce Harrisface)
    How would I go about doing this question? Name:  ImageUploadedByStudent Room1453070079.520693.jpg
Views: 43
Size:  191.2 KB
    Let's get you started. You are given that the observations are independent of one another - therefore to get the combined likelihood, you simply multiply together the likelihoods for each individual observation. They are all Poisson distributed, so we can simply write down the Poisson likelihood for the x's and the y's.

    \displaystyle L(\alpha | x_{i}) = \frac{e^{-\alpha} \alpha^{x_{i}}}{x_{i}!}

    and

    \displaystyle L(\tau | y_{j}) = \frac{e^{-(\tau + \alpha)} (\alpha + \tau)^{y_{j}}}{y_{j}!}

    Now multiply these together for all the  x_{i} and  y_{j} , and you have the combined likelihood as a function of \alpha, \tau

    The second part of the question simply requires you to (a) set  \tau = 0 in this expression and then differentiate w.r.t.  \alpha to get the maximum likelihood estimate. (b) is similar, but messier. Can you now do (c) yourself?
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Break up or unrequited love?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.