## Bitwise-OR operations on random integers

### Problem 323

Published on Sunday, 6th February 2011, 07:00 am; Solved by 1930; Difficulty rating: 20%Let `y`_{0}, `y`_{1}, `y`_{2},... be a sequence of random unsigned 32 bit integers

(i.e. 0 ≤ `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}**|**`y`_{i-1}, for`i`> 0. (**|**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`≥ N.

Find the expected value of N.

Give your answer rounded to 10 digits after the decimal point.