Sequential selection of an increasing subsequence from a sample of random size

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence. We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.