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>
Summary
| Great Lakes | 2025-02-01 | Y | 6 | 11.67 | 83% | 33% | 0% |
| Lower Mid-Atlantic | 2025-02-01 | Y | 1 | 20.00 | 100% | 100% | 0% |
| Midwest | 2025-02-01 | Y | 5 | 22.00 | 100% | 80% | 40% |
| Northeast | 2025-02-01 | Y | 3 | 13.33 | 67% | 67% | 0% |
Data
| Chicago B | Illinois C | 10 | 10 | 10 | 30 |
| Illinois A | Chicago D | 10 | 10 | 0 | 20 |
| WashU A | Indiana A | 10 | 10 | 0 | 20 |
| Chicago A | Indiana B | 10 | 10 | 10 | 30 |
| WashU B | Missoui S&T | 0 | 10 | 0 | 10 |