Using Pearson Correlation to Improve Envelopes around the Distributions of Functions

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Given two random variables whose dependency relationship is unknown, if a new random variable is defined whose samples are some function of samples of the given random variables, the distribution of this function is not fully determined. However, envelopes can be computed that bound the space through which its cumulative distribution function must pass. If those envelopes could be made to bound a smaller space, the cumulative distribution, while still not fully determined, would at least be more constrained. We show how information about the correlation between values of given random variables can lead to better envelopes around the cumulative distribution of a function of their values.

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