Solution
Maximum of Uniforms
Show the problem again
Four values are drawn independently and uniformly at random from [0, 1]. What is the expected value of the maximum? Express as a fraction.
Worked solution
The answer is 4/5. P(max ≤ x) = x⁴, so the density of the maximum is 4x³. Then E[max] = ∫₀¹ x · 4x³ dx = 4/5. In general E[max of n] = n/(n+1): the n draws plus the interval endpoints split [0,1] into roughly equal gaps, leaving 1/(n+1) above the maximum on average.
Source: Standard order-statistics exercise from mathematical statistics texts. Statement written for AxiomIQ.