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Poisson Probability Mass Function & Cumulative Distribution Function

Probability and Statistics

Medium

For a Poisson distribution with rate $\lambda$ , compute the probability of exactly $k$ events:

P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}

Also compute the probability of at most $k$ events:

P(X\le k) = \sum_{i=0}^{k}\frac{e^{-\lambda}\lambda^i}{i!}

Here, $i$ is a possible event count and lam represents $\lambda$ . Return a dictionary containing pmf and cdf as Python floats.

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Examples

Input: lam = 3.0, k = 2

Output: {"pmf": 0.224042, "cdf": 0.42319}

Explanation: The CDF adds the probabilities for zero, one, and two events.

Input: lam = 2.5, k = 0

Output: {"pmf": 0.082085, "cdf": 0.082085}

Input: lam = 1.0, k = 1

Output: {"pmf": 0.367879, "cdf": 0.735759}

Hint 1

Initialize the zero-event probability with math.exp(-lam).

Hint 2

Advance from count i - 1 to i by multiplying the previous probability by lam / i.

Requirements

Start with the probability of zero events
Use the recurrence between consecutive Poisson probabilities
Accumulate probabilities through k for the CDF
Return exactly pmf and cdf in a dictionary of Python floats

Constraints

lam is positive
k is an integer between 0 and 50
Use the Python standard library only

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