Question: [30 points]

Let Y denote the number of heads in n flips of a coin, whose probability of heads is \(\theta \).

1. [15 points] Suppose \(\theta \) follows a distribution \(\Pr \left( \theta \right)=Beta\left( a,b \right)\), and then you observe y heads out of n flips. Show algebraically that the mean \(E\left( \theta |Y=y \right)\) always lies between the mean \(E\left( \theta \right)\) and the observed relative frequency of heads:
   \[\min \left\{ E\left( \theta \right),\frac{y}{n} \right\}\le E\left( \theta |Y=y \right)\le \max \left\{ E\left( \theta \right),\frac{y}{n} \right\}\]
   Here \[E\left( \theta |Y=y \right)\] is the mean of the distribution \(\Pr \left( \theta |Y=y \right)\), and \(E\left( \theta \right)\) is the mean of the distribution \(\Pr \left( \theta \right)=Beta\left( a,b \right)\).
2. [15 points] Show that, if \(\theta \) follows a uniform distribution, \(\Pr \left( \theta \right)=Uni\left( 0,1 \right)\), we have

\(\operatorname{var}\left( \theta |Y=y \right)\le \operatorname{var}\left( \theta \right)\)

Here \(\operatorname{var}\left( \theta |Y=y \right)\) is the variance of the distribution \(\Pr \left( \theta |Y=y \right)\) and \(\operatorname{var}\left( \theta \right)\) is the variance of the distribution \(\Pr \left( \theta \right)=Uni\left( 0,1 \right)\).

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