In this question we will use a basic example to learn about bayesian statistics, which models parameters with prior distributions (sometimes to indicate uncertainty in our beliefs). suppose that you have a coin which may not be fair. its parameter p, which is the chance of landing heads, could in theory lie anywhere in the range [0, 1]. you flip this coin one time and let x draw some conclusions about p. let's choose a prior distribution for this parameter, and assume that p is drawn from a uniform[0, 1] distribution. heads, and now would like to (a) write down a hierarchical model for this scenario, which should be of the form (some variable) ~(some distribution) (some other variable) | (the first variable) ~(some distribution) (b) now calculate the following: for any t e [0, 1], find p(p < t, x 0) and p(p t, x 1). to do these calculations, you are working with a joint distribution where one variable is discrete and one is continuous; intuitive rules will apply for combining integrals and sums, etc. for example it's fine to write p(x = 1 | p = p) = p. (c) finally calculate the conditional distribution of p, given that you observe x = 1. to do this, start with the conditional cdf, i. e. p(pt|x 1), then get the density. in bayesian statistics, the conditional distribution of p given our observed value of x, is called the posterior distribution for p-meaning its distribution after observing the data (which in this case is x, the data from tossing the coin)