import math

'''
Problem 1
'''
def normal_cdf(x, mu=0,sigma=1):
return (1 + math.erf((x - mu) / math.sqrt(2) / sigma)) / 2

def inverse_normal_cdf(p, mu=0, sigma=1, tolerance=0.00001):
"""find approximate inverse using binary search"""

# if not standard, compute standard and rescale
if mu != 0 or sigma != 1:
   return mu + sigma * inverse_normal_cdf(p, tolerance=tolerance)

low_z, low_p = -10.0, 0          # normal_cdf(-10) is (very close to) 0
hi_z, hi_p = 10.0, 1          # normal_cdf(10) is (very close to) 1
while hi_z - low_z > tolerance:
   mid_z = (low_z + hi_z) / 2 # consider the midpoint
   mid_p = normal_cdf(mid_z)    # and the cdf's value there
   if mid_p < p:
         # midpoint is still too low, search above it
         low_z, low_p = mid_z, mid_p
   elif mid_p > p:
         # midpoint is still too high, search below it
         hi_z, hi_p = mid_z, mid_p
   else:
         break

return mid_z

def normal_approximation_to_binomial(n, p):
"""finds mu and sigma corresponding to a Binomial(n, p)"""
mu = p * n
sigma = math.sqrt(p * (1 - p) * n)
return mu, sigma

def normal_upper_bound(probability, mu=0, sigma=1):
"""returns the z for which P(Z <= z) = probability"""
return inverse_normal_cdf(probability, mu, sigma)

def normal_lower_bound(probability, mu=0, sigma=1):
"""returns the z for which P(Z >= z) = probability"""
return inverse_normal_cdf(1 - probability, mu, sigma)

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