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Area between the CDFs

Expected valueEasy

Strategies $X$ and $Y$ both pay $0$ , $20$ or $50$ points. $X$ has chances $20%$ , $30%$ and $50%$ . $Y$ has chances $40%$ , $30%$ and $30%$ . $X$ first-order dominates $Y$ . What is the area between their distribution functions, the integral of $F_{Y} (t) - F_{X} (t)$ over all $t$ ? Give a whole number.

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Hint

Find the gap between the curves on each stretch between payoffs, then multiply by its length.

Solution

From $0$ to $20$ the cumulative chances are $0.2$ for $X$ and $0.4$ for $Y$ , a gap of $0.4 - 0.2 = 0.2$ .
From $20$ to $50$ they are $0.5$ and $0.7$ , a gap of $0.7 - 0.5 = 0.2$ .
As a check, the means are $0.3 \times 20 + 0.5 \times 50 = 31$ and $0.3 \times 20 + 0.3 \times 50 = 21$ , and their difference equals the area.
The area is $0.2 \times 20 + 0.2 \times 30 = 10$ .

Answer: $10$

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