What is probability of a head in a single toss of the coin

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1. One of my prized possessions is my 'simplified quantum coin', described by P(X = 1|p) = p, P(X =0|p) = 1 - p, where X is the number of heads in a toss of the coin, and p is a random variable with P(p = 1/4) = 1/4 and P(p = 3/4) = 3/4.

(a) What is the probability of a 'head' in a single toss of this coin?

(b) If the coin is tossed and a 'head' results, what is the probability that p is 3/4?

2. Consider a random sample X₁, , X_n from a Uniform distribution on (0, θ). Derive a likelihood ratio test of H₀ : θ = θ_o versus H_A : θ ≠ θ₀. Show that the rejection region is of the form X_(n) ≤ cθ_o or X_(n) > θ_o and find c for level α = 0.05.

Reference no: EM131025693

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