Question

Answer the following about difficulties that arise when adding two random variables, for 10 points each.
[10m] If they are independent and each has a density, the density of their sum equals this operation of their densities. For functions f and g, this operation is denoted “f star g” and defined via an integral of their shifted product.
ANSWER: convolution [or word forms like convolve]
[10e] Conveniently, this quantity is linear under any dependence between the random variables. This quantity, which is a distribution’s first moment, is typically denoted “mu” or “X-bar.”
ANSWER: expected value [or EV; accept expectation value; accept arithmetic mean or average]
[10h] To the bane of many intro probability students, variance is not linear in general. For arbitrary random variables X and Y, the variance of “X plus Y” equals the variance of X, plus the variance of Y, plus what expression?
ANSWER: two times the covariance of X and Y [accept equivalents like twice the covariance; reject “covariance” or other answers that omit the factor of two]
<TM, Other Science>

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Summary

Great Lakes2025-02-01Y611.6783%33%0%
Lower Mid-Atlantic2025-02-01Y120.00100%100%0%
Midwest2025-02-01Y522.00100%80%40%
Northeast2025-02-01Y313.3367%67%0%

Data

Carnegie Mellon ACarnegie Mellon B1010020
Michigan ACase Western A1010020
Michigan BOhio State A010010
Michigan CCase Western B010010
Michigan DKenyon0000
Ohio State BMichigan State010010
Virginia Tech AVirginia A1010020
Chicago BIllinois C10101030
Illinois AChicago D1010020
WashU AIndiana A1010020
Chicago AIndiana B10101030
WashU BMissoui S&T010010
Brandeis BHarvard A0000
Harvard BBrandeis A1010020
MIT ABrown A1010020