For an arbitrary random variable x

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For an arbitrary random variable X, use the Chebyshev inequality to show that the probability that X is more than k standard deviations from its expected value E[X] satisfies

For a Gaussian random variable Y, use the ?(·) function to calculate the probability that Y is more than k standard deviations from its expected value E[Y]. Compare the result to the upper bound based on the Chebyshev inequality.

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