Question: [20 points]

Suppose that Y is binomially distributed \(Y\tilde{\ }Bin\left( n=5,\theta \right)\) where \(\theta \) is unknown. Furthermore, assume that

and that a priori we view each of these 11 possibilities as equally likely, i.e.

1. [10 points] Using the Bayes’ rule, write down a formula for the posterior distribution \(\Pr \left( \theta |y \right)\) in terms of \({{\theta }_{i}}\), and simplify as much as possible.
2. [5 points] For y = 0, make a plot for \(\Pr \left( \theta |y \right)\) for \(\theta \in \Theta =\left\{ 0.0,0.1,0.2,...,0.9,1.0 \right\}\). In
   other words, make a plot with the horizontal axis representing the 11 values of \(\theta \) and the vertical axis representing the corresponding values of \(\Pr \left( \theta |y \right)\).
3. [5 points] Repeat (b) for each \(y\in \left\{ 1,2,3,4,5 \right\}\) so in the end you have six plots (including the one in (b)). Describe what you see in your plots and discuss whether or not they make sense.

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