In example [3.24] of the binary random process formed by taking the binary expansion of a…

In example [3.24] of the binary random process formed by taking the binary expansion of a uniformly distributed number on [0, 1], find the pmf for the random variable Xn for a fixed n. Find the pmf for the random vector (Xn, Xk) for fixed n and k. Consider both the cases where n = k and where n ≠ k. Find the probability Pr(X5 = X12).

Example [3.24]

Consider the probability space of example [3.23], but cut it down to the unit interval; that is, consider the probability space ([0, 1), B([0, 1)), P) where P is the probability measure induced by the pdf f(r) = 1; r ∈ [0, 1). (So far this is just another model for the same thing.) Define for n = 1, 2 … ,Xn(ω) = bn(ω) = the nth digit binary expansion of ω, that is;

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