I am trying to see what would be an appropriate prior distribution for $\theta$, the average GPA of all college students who graduate in a particular year Y from university U. Assume also that from past data we know that usually about $50\%$ of graduating college students from U obtain a grade of $3.0/4.0$ or higher.
What would be a meaningful prior distribution for $\theta$? Given that we are not completely clueless about the past distribution of $\theta$, it would be better not to put a uniform prior on $\theta$. I was thinking of putting a normal prior distribution on $\theta$ but this seems problematic too, since the support of a normal distribution is $\Bbb R$ while the support of the prior for $\theta$ is $[0,4]$. IS there some natural choice for the prior distribution given the above?