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February 13th, 2018, 02:49 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  Posterior Distribution from Beta Density with Exponential Prior
Let $X_1,...,X_n$ be iid random variables with a common density function given by: $f(x\theta)=\theta x^{\theta1}$ for $x\in[0,1]$ and $\theta>0$. Put a prior distribution on $\theta$ which is $EXP(2)$, where $2$ is the mean of the exponential distribution. Obtain the posterior density function of $\theta$. Do you recognize this distribution? Clearly $f(x\theta)$ is a beta distribution. So this implies that the posterior distribution is $$f(\thetax)\propto f(x\theta)\pi(\theta)=\frac{\theta x^{\theta1}e^{\theta/2}}{2}$$ But I don't really recognize the distribution at all. Was my calculation incorrect? If not, what is the name of this distribution? 
February 13th, 2018, 08:35 PM  #2 
Senior Member Joined: Oct 2009 Posts: 882 Thanks: 339 
Looks like a gamma.

February 14th, 2018, 08:41 AM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,574 Thanks: 1422 
Was it you that got this answered over at Stack Exchange? Are you all set now?

February 14th, 2018, 12:28 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
posterior = ass ?

February 14th, 2018, 01:04 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,574 Thanks: 1422  

Tags 
beta, density, distribution, exponential, posterior, prior 
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