Overall question: Assume X$_i$ ~ i.i.d. Unif(0,1). What is the expected length of a sequence that is monotonically increasing when drawn from the distribution?
The skeleton of the solution is
$$ \sum_n n*prob(length=n) = \sum_n 1/n! = e $$
How to compute Prob(length = 2)?
Can Prob(length = 2) be worded as: Suppose we are drawing iid random variables from a uniform[0,1] distribution. What is the probability that the first 3 draws X1, X2, X3 are such that $$ X1 < X2 \space and \space X3<X2 \space ? $$
Or does prob(length = 2) mean simply $$ Prob(X1<X2)? $$ I'm looking for a solution using integrals...