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Convergence in Probability

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A. If g is a real-valued function that is continuous at a, then g(a^) converges in probability to g(a).
b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0 < y <1, 0 elsewhere. Show that converges in probability to some constant and find the constant.

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The solution contains the proof of the theorem "If g is a real-valued function that is continuous at a, then g(a^) converges in probability to g(a)" and an application of this theorem for a continuous random variable.

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1. Proofs
a. Prove Theorem 9.2(d): If g is a real-valued function that is continuous at , then converges in probability to .
b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0  ...

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