from math import pow, exp, factorial

class Poisson:

    def __init__(self, rate):
        self.rate = rate

    def prob_exactly(self, n, t):
        rate = self.rate * t
        return pow(rate, n) * exp(-rate) / factorial(n)

    def prob_at_least(self, n, t):
        complements = range(n)
        total = 0.0

        for c in complements:
            p = self.prob_exactly(c, t)
            total += p

        return 1 - total

    def prob_at_most(self, n, t):
        return 1 - self.prob_at_least(n + 1, t)

pois = Poisson(2)

pois.prob_exactly(0, 2)

pois.prob_at_least(2, 3)

class Exponential:

    def __init__(self, rate):
        self.rate = rate

    def prob_less_than_or_equal(self, t):
        rate = self.rate * t
        return 1 - exp(-rate)

    def prob_greater_than(self, t):
        return 1 - self.prob_less_than_or_equal(t)

    def prob_between(self, t1, t2):
        p1 = self.prob_less_than_or_equal(t1)
        p2 = self.prob_less_than_or_equal(t2)

        return p2 - p1

expo = Exponential(2)

expo.prob_less_than_or_equal(0.25)

expo.prob_between(0.25, 0.5)

expo.prob_greater_than(1.5)

pois.prob_exactly(0, 1.5)