Problems

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Poisson Probability Mass Function & Cumulative Distribution Function

Probability and Statistics

Medium

Implement the Poisson distribution Probability Mass Function (PMF) and Cumulative Distribution Function (CDF). The Poisson distribution models the count of events in a fixed interval with average rate λ.

Poisson Distribution:

Probability Mass Function:

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

Cumulative Distribution Function:

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

Function Arguments

lam: float - Rate parameter (λ > 0)
k: int - Number of events (k ≥ 0)

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Examples

Input: lam=3, k=2

Output: pmf≈0.2240, cdf≈0.4232

Input: lam=2.5, k=0

Output: pmf≈0.0821, cdf≈0.0821

Input: lam=1.0, k=1

Output: pmf≈0.3679, cdf≈0.7358

Hint 1

Use np.log() and np.exp() for numerical stability in factorial computation.

Hint 2

For CDF, sum PMF values from i=0 to k using a loop.

Hint 3

Use np.sum(np.log(np.arange(1, k+1))) for log factorial.

Requirements

Return tuple: (pmf, cdf)
Both pmf and cdf: scalar floats
Use stable log factorial computation
No loops over factorial calculation

Constraints

λ > 0, k ≥ 0 ≤ 50
NumPy only; time limit: 300ms

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