4. (10 points) Consider the following experiment. We draw a real number p from the range [0,1] with uniform probability,

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4 10 Points Consider The Following Experiment We Draw A Real Number P From The Range 0 1 With Uniform Probability 1 (99 KiB) Viewed 31 times

4. (10 points) Consider the following experiment. We draw a real number p from the range [0,1] with uniform probability, and we let the value p be the probability of getting Heads when we toss a biased coin. We toss the biased coin a total of n times. (a) Show that for an integer values of n and y, where 0 <y<n, we have: 1 dx x'(1 - 1)"-y = nto 1 n y = 0,1,...,n (1) n+1 Hint: this requires repeated applications of integration by parts. 2 (b) Show that the probability of getting y Heads in the n tosses is nt1, which is independent of the value of y. (c) Given that you observe y Heads, what is the probability density function for the bias of the coin?