Bitwise-OR operations on random integers

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Problem 323

Let y0, y1, y2,... be a sequence of random unsigned 32 bit integers
(i.e. 0 ≤ yi < 232, every value equally likely).

For the sequence xi the following recursion is given:

  • x0 = 0 and
  • xi = xi-1| yi-1, for i > 0. ( | is the bitwise-OR operator)

It can be seen that eventually there will be an index N such that xi = 232 -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.