Program to find expected value of given equation for random numbers in Python

Suppose we have a number n. Consider x = rand() mod n, where rand() function generates integers between 0 and 10^100 (both inclusive) uniformly at random. And

$$Y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$

We have to find the expected value of Y. The value of n will be in range 1 and 5*10^6.

So, if the input is like n = 5, then the output will be 1.696

To solve this, we will follow these steps −

• err := 2235.023971557617
• max_n := 5 * 10^6
• pref := a list initially contains a single 0
• for i in range 1 to 5 * 10^6, do
  • insert (last item of pref + (1 +(4*i + 1)^0.5) * 0.5 at the end of pref
• if n
• return pref[n - 1] / n

Example

Let us see the following implementation to get better understanding −

def solve(n):
   err = 2235.023971557617
   max_n = 5 * 10**6

   pref = [0]
   for i in range(1, 5 * 10**6):
      pref.append(pref[-1] + (1 + (4 * i + 1)**0.5) * 0.5)

   if n 
Input
5
Output
1.69647248786