(Solution) - Consider the probability space A P and let A1 -(2025 Original AI-Free Solution)

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Consider the probability space ((, A, P), and let A1,..., An be independent events with P(Ak) = p, k = 1, ..., n. Next, define the function X : ( ? ( as follows:
X(() = the number of A1,..., An containing (.
Then show that
(i) X is a r.v.
(ii) The distribution of X is B(n, p) (i.e., Binomial with parameters n and p)?