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Finding maximum likelihood function for 2 normally distributed samples

  1. Apr 2, 2012 #1
    1. The problem statement, all variables and given/known data
    The question is about how to combine to different samples done with 2 different methods of the same phenomena.

    Method 1 gives normally distributed variables [itex]X_1,X_2,...X_{n_1}[/itex], with [itex]\mu[/itex] and [itex]\sigma^2_1[/itex]

    Method 1 gives normally distributed variables [itex]Y_1,Y_2,...,Y_{n_2}[/itex], with [itex]\mu[/itex] and [itex]\sigma^2_2[/itex]

    All measurements can be considered independent.

    Define [itex]\overline{X}=1/n_1*\Sigma^{n_1}_{i=1}X_i[/itex] and [itex]\overline{Y}={1/n_2}*\Sigma^{n_2}_{j=1}Y_j[/itex]

    Both the samples are taken from the same population so they have the same [itex]\mu[/itex]


    The question is set up a likelihood function for the [itex]n_1 + n_2[/itex] measurements, and show that the maximum likelihood estimator is given by

    [itex]\mu_{mle}={(\sigma^2_2n_1\overline{X}+\sigma_1^2n_2\overline{Y})/(\sigma_2^2n_1+\sigma^2_1n_1)}[/itex]

    2. Relevant equations

    The likelihood function for 1 sample of a normally distributed function.

    [itex]L(\mu)=L(X_1,X_2,...X_n;\mu,\sigma^2)=1/((2\pi\sigma^2)^{n/2})*e^{-1/2*\Sigma^{n}_{i=1}(X_i-\mu)/\sigma}[/itex]

    3. The attempt at a solution

    The main problem is that I don't know how I should handle the 2 samples instead of 1, I'm not sure how those two should combine.


    Edit; I think I came up with the solution while I was writing up this which took quite some time since I don't know latex. I'll see when I try it out.

    Since the measurement is presumed independent does the likelihood function become something like?

    [itex]L(\mu)=L(X_1,X_2,...X_{n_1};\mu,\sigma^2_1)*L(Y_1,Y_2,...,Y_{n_2};\mu,\sigma_2^2)[/itex]
     
    Last edited: Apr 2, 2012
  2. jcsd
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