What is the expected length of a sequence that is monotonically increasing when drawn from the distribution?

First thing is to phrase the question in the notation of a probability. We are looking for - p(x1<x2,x3<x2) Next step is to simply expand this based on the rules of conditional probability as - p(x1<x2,x3<x2)=â«10p(x1<x2,x3<x2|x2)p(x2)dx2 Given we are working with uniform[0,1] variables, the first term can be written simply - p(x1<x2,x3<x2|x2)=x22 We also know that p(x2)=1 on the interval we care about. Therefore we have - p(x1<x2,x3<x2)=â«10x22dx2=x33âŁâŁâŁ10=1/3