Syntax error Program to find expected value of maximum occurred frequency values of expression results in Python

Program to find expected value of maximum occurred frequency values of expression results in Python

Suppose we have M different expressions, and the answers of these expressions are in range 1 to N (both inclusive) So consider x = max(f(i)) for each i in range 1 through N, we have to find the expected value of x.

So, if the input is like M = 3, N = 3, then the output will be 2.2, because

Sequence Maximum frequency
111 3
112 2
113 2
122 2
123 1
133 1
222 3
223 2
233 2
333 3

$$E(x) = \sum P(x) * x = P(1) + 2P(2) + 3P(3) = \frac{1}{10} + 2 * \frac{6}{10} + 3 * \frac{3}{10} = \frac{22}{10}$$

To solve this, we will follow these steps −

  • combination := a new map
  • Define a function nCr() . This will take n, k_in
  • k := minimum of k_in and (n - k_in)
  • if n < k or k < 0, then
    • return 0
  • otherwise when (n, k) is in combination, then
    • return combination[n, k]
  • otherwise when k is same as 0, then
    • return 1
  • otherwise when n is same as k, then
    • return 1
  • otherwise,
    • a := 1
    • for cnt in range 0 to k - 1, do
      • a := a * (n - cnt)
      • a := floor of a/(cnt + 1)
      • combination[n, cnt + 1] := a
    • return a
  • From the main method, do the following:
  • arr := a new list
  • for k in range 2 to M + 1, do
    • a := 1
    • s := 0
    • for i in range 0 to floor of M/k + 2, do
      • if M < i * k, then
        • come out from loop
      • s := s + a * nCr(N, i) * nCr(N-1+M-i*k, M-i*k)
      • a := -a
    • insert s at the end of arr
  • total := last element of arr
  • diff := an array where insert arr[0] at beginning then add a list where (arr[cnt + 1] - arr[cnt]) for each cnt in range 0 to M - 2
  • output := sum of all elements present in (diff[cnt] *(cnt + 1) / total for cnt in range 0 to M-1)
  • return output

Example

Let us see the following implementation to get better understanding −

combination = {}
def nCr(n, k_in):
   k = min(k_in, n - k_in)
   if n < k or k < 0:
      return 0
   elif (n, k) in combination:
      return combination[(n, k)]
   elif k == 0:
      return 1
   elif n == k:
      return 1
   else:
      a = 1
      for cnt in range(k):
         a *= (n - cnt)
         a //= (cnt + 1)
         combination[(n, cnt + 1)] = a
      return a

def solve(M, N):
   arr = []
   for k in range(2, M + 2):
      a = 1
      s = 0
      for i in range(M // k + 2):
         if (M < i * k):
            break
         s += a * nCr(N, i) * nCr(N - 1 + M - i * k, M - i * k)
         a *= -1
      arr.append(s)
   total = arr[-1]
   diff = [arr[0]] + [arr[cnt + 1] - arr[cnt] for cnt in range(M - 1)]
   output = sum(diff[cnt] * (cnt + 1) / total for cnt in range(M))
   return output

M = 3
N = 3
print(solve(M, N))

Input

3, 3

Output

1
Updated on: 2021-10-11T09:20:44+05:30

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