Suppose P is drawn from the Uniform[0, 1] distribution, and then conditional on P, another random variable X is drawn from a Bernoulli(P) distribution. (a) Use the tower law to compute P(P ≤ t, X = 1) where t ∈ [0, 1] is some constant value

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Suppose P is drawn from the Uniform[0, 1] distribution, and then conditional on P, another random
variable X is drawn from a Bernoulli(P) distribution.
(a) Use the tower law to compute P(P ≤ t, X = 1) where t ∈ [0, 1] is some constant value.
(b) Compute the CDF of the conditional distribution of P given X = 1.
(c) Now suppose that, after P is drawn, we instead draw twice from the Bernoulli(P) distribution
to produce random variables X and Y , rather than drawing only once to produce X as before.
(We assume that, conditional on the value of P, the random variables X and Y are drawn
independently.) Calculate P(X ≠Y ).

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