Expected first value given two uniforms sum to 0.8

Question

You pick two numbers X and Y independently, each uniform on the interval [0, 1]. You are told that X + Y = 0.8. Given that X + Y = 0.8, what is the expected value of X?

Accepted Answer

You want E[X \mid X + Y = 0.8]. Even though these are continuous numbers, the symmetry argument is identical to the discrete case. X and Y come from the same distribution and enter the sum in identical roles, so they share the same conditional expectation given the sum: E[X \mid X + Y = 0.8] = E[Y \mid X + Y = 0.8]. The two conditional expectations must add to the known sum: E[X \mid X + Y = 0.8] + E[Y \mid X + Y = 0.8] = 0.8. Two equal numbers adding to 0.8 are each 0.4, so E[X \mid X + Y = 0.8] = 0.4. (You can also see it geometrically: given the sum is 0.8, the point (X, Y) is uniform alon…

Expected first value given two uniforms sum to 0.8

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