Warming Spells

A weather station records daily high temperatures $T_1, \dots, T_n$ on $n$ consecutive days, modelled as i.i.d. draws from a continuous distribution (so ties have probability zero). A warming spell starts on day $i$ if $T_i < T_{i+1} < T_{i+2} < T_{i+3} < T_{i+4}$. Spells may overlap. Find the value of $n$ for which the expected number of warming spells is exactly 2.