This way we say that we generate a random variable x

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Unformatted text preview: f values for X . This way we say that we generate a random variable X . Let FX (x ) be the CDF of X . From the description, we have X = F −1 (U ) and U = F (X ). We have FX (x ) = P (X ≤ x ) = P F −1 (U ) ≤ x = P (U ≤ F (x )) = F (x ) Thus the generated random variable X follows the desired CDF F (x ). M. Chen (IE@CUHK) ENGG2430C lecture 10 3 / 29 Inequality: Motivation In engineering design problems, often we care about bounding the probability of a “bad” event. In wireless communication: bound the probability of error decoding. In ATV poll problem: bound the probability of our estimation being too off. But computing the exact probability of the events can be complicated. Sometime critical information for their computation is not available. What can we do? Essentially, computing probability exactly needs distribution functions, which can be hard to get. One way out is to compute a bound on the probability using less information, e.g., mean and variance. M. Chen (IE@CUHK) ENGG2430C lecture 10 4 / 29 Markov Inequality If a random variable X takes only nonnegative values, then P (X...
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