The __________ distribution has the following equations:

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The question states: 'The __________ distribution has the following equations:' and provides the variance formula Var(X) = σ^2 = (b − a)^2 / 12. This specific variance expression is a hallmark of the Uniform distribution on the interval [a, b], whose standard result is Var(X) = (b − a)^2 / 12. In contrast, many other common distributions have different variance formulas that depend on other parameters (for example, Normal distributions involve variance σ^2 not in the (b−a) form, and Exponential/Geometric have variances tied to rate parameters rather than an interval width). Therefore, the variance equation given aligns with a Uniform distribution over [a, b].

Since the answer options are not provided here, there is no direct alternative to compare within the list. However, understanding that (b − a)^2 / 12 is the variance for Uniform(a, b) helps identify the blank as referring to the Uniform distribution. Keep in mind that if other distributions were listed, you would compare their standard variance expressions to see which one matches (b − a)^2 / 12, anditis unlikely to match for distributions that do not have a finite support width parameterized by a and b in the same way.

In summary, recognizing that the interval width (b − a) governs the spread for a Uniform distribution explains why its variance takes the form (b − a)^2 / 12, distinguishing it from other common distributions' variance formulas.

ECON 3640-090 Spring 2025 Lecture #16 Warm-Up

The __________ distribution has the following equations:

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