Compute the total variation distance between \( \mathbf{P}=\operatorname{Ber}(p) \quad \) and \( \quad \mathbf{Q}=\operatorname{Ber}(q), \quad \) where \( p, q \in[0,1] \) (If applicable, enter \( \operatorname{abs}(x) \) for \( |x| \).) \( \operatorname{TV}(\mathbf{P}, \mathbf{Q})= \) Let \( X_{1}, \ldots, X_{n} \) be \( n \) i.i.d. Bernoulli random variables with some parameter \( p \in[0,1] \), and \( \bar{X}_{n} \) be their empirical average. Consider the total variation distance \( \operatorname{TV}\left(\operatorname{Ber}\left(\bar{X}_{n}\right), \operatorname{Ber}(p)\right) \) between \( \operatorname{Ber}\left(\bar{X}_{n}\right) \) and \( \operatorname{Ber}(p) \) as a function of the random variable \( \bar{X}_{n} \), and hence a random variable itself. Does \( \operatorname{TV}\left(\operatorname{Ber}\left(\bar{X}_{n}\right), \operatorname{Ber}(p)\right) \) necessarily converge in probability to a constant? If yes, enter the constant below; if not; enter DNE. \[ \operatorname{TV}\left(\operatorname{Ber}\left(\bar{X}_{n}\right), \operatorname{Ber}(p)\right) \stackrel{(\mathrm{P})}{\longrightarrow} \] STANDARD NOTATION
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Recall that the total variation distance between two probability distributions \(P\) and \(Q\) is defined as: \(\operatorname{TV}(P, Q) = \frac{1}{2} \sum_{x \in \mathcal{X}} |P(x) - Q(x)|\) For Bernoulli distributions, the sample space \(\mathcal{X}\) consists Show more…
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