Solution
A Clock and a Fuse
Show the problem again
X is exponentially distributed with mean 1, and U is uniform on [0, 1], independent of X. What is E[min(X, U)]? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is 1/e ≈ 0.3679. For a non-negative random variable, E[M] = ∫₀^∞ P(M > t) dt. Here P(min(X, U) > t) = P(X > t)·P(U > t) = e^(−t)(1 − t) for 0 ≤ t ≤ 1, and 0 afterwards. So E[min(X, U)] = ∫₀¹ (1 − t) e^(−t) dt. Integrating by parts, ∫₀¹ e^(−t) dt = 1 − 1/e and ∫₀¹ t e^(−t) dt = 1 − 2/e, so the expectation is (1 − 1/e) − (1 − 2/e) = 1/e.
Source: Original problem written for AxiomIQ.