Let $y_0,y_1,y_2,\dots$ be a sequence of random unsigned $32$-bit integers
(i.e. $0\le y_i<2^{32}$, every value equally likely).

For the sequence $x_i$ the following recursion is given:

• $x_0=0$ and
• $x_i=x_{i-1}\mathbf{\mid }{y}_{i-1}$, for $i>0$. ($\mathbf{\mid }$ is the bitwise-OR operator).

It can be seen that eventually there will be an index $N$ such that $x_i=2^{32}-1$ (a bit-pattern of all ones) for all $i\ge N$.

Find the expected value of $N$.
Give your answer rounded to $10$ digits after the decimal point.