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Suppose that $X_1,\ldots,X_n$ is a random sample from $U(θ,θ+1),$ and further assume $θ$ has a prior distribution as the discrete uniform distribution on the integers $1,2,\ldots,n$ where $n$ is known. Obtain the posterior distribution of $θ.$

The problem is that as the joint distribution has an indicator function, then I don't know how to integrate the product of this times $1/n$ (the prior distribution).

Any one could help me with that?

Thank you in advance.

Michael Hardy
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MGF01
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  • Don't we have $\theta=\lfloor X_1\rfloor$ here? And, if so, is that enough to solve your problem? I am not really familiar with the concept "prior-posterior", but I was just thinking. – drhab Mar 08 '18 at 13:02
  • No, We haven't. The prior distribution is just the distribution of the parameter $\theta$ – MGF01 Mar 08 '18 at 17:01
  • Right, but if $x_i \in [i,i+1]$ for some $i$, then it can only be such that $\theta = i$. This means your posterior distribution would be a point mass at $\delta_i$. – Ryan Warnick Mar 08 '18 at 17:52
  • I don't really understand it – MGF01 Mar 08 '18 at 18:36
  • Now I understand it, thank you to both of you. You two are right! – MGF01 Mar 08 '18 at 20:38

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