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

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 < max_n, then
    • return pref[n - 1] / n
  • otherwise,
    • total :=(4 *(n - 1) + 5)^1.5 / 6 - 5^1.5 / 6 - err
    • ans := 0.5 + total /(2 * n)
    • return ans

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 < max_n:
      return pref[n - 1] / n
   else:
      total = (4 * (n - 1) + 5)**1.5 / 6 - 5**1.5 / 6 - err
      ans = 0.5 + total / (2 * n)
      return ans

n = 5
print(solve(n))

Input

5

Output

1.69647248786
Updated on: 2021-10-11T07:38:29+05:30

274 Views

Kickstart Your Career

Get certified by completing the course

Get Started
Advertisements