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Q1. (8 pts)

The probability density function of the beta distribution can be written as: \[p(x|\alpha, \beta) = x^{\alpha -1} (1-x)^{\beta -1} \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)},\] for \(x \in [0,1]\).

Plot the density function for a beta distribution with:

How do the values of \(\alpha\) and \(\beta\) change the distribution?

Q2. (8 pts)

Simulate 1000 observations from each of the distributions below:

Use the samples from each distribution to calculate the mean and variance. How do your calculations compare with the known mean and variance from these distributions?

Q3. (4 pts)

A main emphasis in Bayesian statistics is to think about the entire distribution of possibilities, rather than a point interval.