Answer: Graph D

The question asks which of the four distributions (A, B, C, D) has the **highest variance**, given that all random variables are supported on [-10, 10] and we're looking at their CDFs with the mean marked by the vertical dotted line. Variance measures **spread** around the mean. For variables restricted to the same interval [-10, 10], the distribution with the **highest variance** is the one that puts as much probability mass as possible at the **extreme ends** (−10 and +10). Let's quickly characterize each graph: - **A**: Very sharp S-shaped CDF, almost all the rise happens in a very narrow region around the mean (≈ 0 to ≈ 3). → Very concentrated around the mean → **low variance** - **B**: Still quite steep / sigmoidal, but noticeably wider than A (rise starts clearly left of 0 and finishes right of 0). → More spread out than A → **moderate variance** - **C**: Much flatter S-curve. It starts rising noticeably before x = −5, reaches ≈ 0.5 near x = 0, and continues rising significantly past x = 5. → Much more spread out → **high variance** - **D**: Perfectly straight line from (−10, 0.2) to (10, 0.8) → this is (essentially) a **uniform distribution** on roughly [−10, 10] (possibly shifted/truncated slightly at the ends, but very close to Uniform[−10, 10]). → Maximum possible spread within the interval → **highest variance** ### Quick ranking of variance (highest to lowest) D > C > B > A ### Why D has the highest variance Among all distributions supported on [a, b], the **maximum possible variance** is achieved by the **two-point distribution** that puts 50% mass at a and 50% mass at b (variance = (b−a)²/4). The **uniform distribution** on [a, b] has variance (b−a)²/12 ≈ 0.083(b−a)², which is the second highest among all continuous distributions — and very close to the theoretical maximum. Graph D is (very nearly) uniform on [−10, 10], while the others are visibly more concentrated toward the center → D has the largest spread → **highest variance**. **Answer: D**

Author: LeetQuiz .