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# hw3 - both 1 2 Find the most probable number of boys and...

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Homework 3 Probability Theory (MATH 235A, Fall 2007) 1. Zero-one law. Let A 1 ,A 2 ,... be independent events. Prove that the events lim sup n A n and lim inf n A n each have probabilities either zero or one. 2. Bounding random variables. Let X be a random variable. Prove that for every ε > 0 there exists K such that P ( | X | > K ) < ε. (Hint: Using Theorem 3.2, show that the probabilities of the events A n = {| X | > n } converge to zero as n → ∞ .) 3. Approximation by bounded random variables. Let X be a random variable. Show that for every ε > 0 there exists a bounded random variable X ε such that P ( X negationslash = X ε ) < ε. (Hint: use Problem 2). 4. Absolute value Show that if X is a random variable then so is | X | . 5. Family There are 4 children in a family. The probabilities of a boy or a girl are
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Unformatted text preview: both 1 / 2. Find the most probable number of boys and girls in the family, and the corresponding probability. 6. Composition of measurable maps. Prove that a composition of two measurable maps is measurable. 7. Random variables induce probability measures on R Let X be a random variable on a probability space (Ω , F , P ). Prove that X induces a probability measure on R in the following sense. For every Borel subset A of R , de±ne P ( A ) := P ( X ∈ A ) = P ( ω ∈ Ω : X ( ω ) ∈ A ) . Prove that ( R , R ,P ) is a probability space....
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