Poisson Waiting Time Probability

Problem

Probability that the k-th event of a Poisson process with rate lam has occurred by time t, which is the probability of at least k events in [0, t]:

P(T_k <= t) = 1 - sum from i = 0 to k - 1 of e^(-lam t) (lam t)^i / i!.

Return 1.0 for k <= 0 and 0.0 for negative rates or times.

Hint

The waiting time is Erlang distributed; use the Poisson count complement.

case 1

in: [1,1,1]

expected: 0.6321205588285577

case 2

in: [2,3,2]

expected: 0.9826487347633355

case 3

in: [1,1,0]

expected: 1

case 4

in: [1,1,3]

expected: 0.08030139707139416

case 5

in: [1,-1,1]

expected: 0

Starter codePython

import math


def poisson_waiting_time_probability(lam, t, k):
    # Your code here
    pass

5 test cases · run your code against them in the editor