19 / 25 Distributions

The Last Value of a Decreasing Run

Draw X1,X2,…X_1, X_2, \dots iid Uniform(0,1). Let NN be the first index at which the running minimum fails to update: N=min⁡{n:Xn≠min⁡(X1,…,Xn)}.N = \min{n : X_n \neq \min(X_1, \dots, X_n)}. So the draws fall, X1>X2>⋯>XN−1X_1 > X_2 > \dots > X_{N-1}, and then XN>XN−1X_N > X_{N-1} breaks the run.

What is E[XN−1]E[X_{N-1}], the expected value of the smallest draw seen?

A

1/21/2

B

3−e≈0.2823 - e \approx 0.282

C

e−2≈0.718e - 2 \approx 0.718

D

1/3≈0.3331/3 \approx 0.333