Distributions

Poisson Probability

Compute P(X=k) for a Poisson(λ) distribution: the probability of exactly k events in a fixed interval given mean rate λ. Formula: (λ^k × e^−λ) / k!. Three named pieces: lam_k = λ^k, e_neg_lam = math.exp(−λ), k_fact = math.factorial(k). Assemble as numerator / k_fact. Library: scipy.stats.poisson.pmf(k, lam).

By hand

λ=3, k=2. Three pieces: lam_k = 3²=9, e_neg_lam = e^−3 ≈ 0.049787, k_fact = 2!=2. Numerator = 9 × 0.049787 ≈ 0.447884. Divide by 2: prob ≈ 0.224042. Each intermediate is visible before the final division.

naive.py
Replay: real traced execution (multi-file project)
import math
lam = 3
k = 2
lam_k = lam ** k
e_neg_lam = math.exp(-lam)
k_fact = math.factorial(k)
numerator = lam_k * e_neg_lam
prob = numerator / k_fact
print('RESULT:', round(prob, 6))

  1. import math

    1import math2lam = 3

  2. lam ← 3

    1import math2lam = 33k = 2
    values this step3lam

  3. k ← 2

    2lam = 33k = 24lam_k = lam ** k
    values this step2k

  4. lam_k ← 9

    3k = 24lam_k = lam ** k5e_neg_lam = math.exp(-lam)
    values this step9lam_k

  5. e_neg_lam ← 0.049787068367863944

    4lam_k = lam ** k5e_neg_lam = math.exp(-lam)6k_fact = math.factorial(k)
    values this step0.049787068367863944e_neg_lam

  6. k_fact ← 2

    5e_neg_lam = math.exp(-lam)6k_fact = math.factorial(k)7numerator = lam_k * e_neg_lam
    values this step2k_fact

  7. numerator ← 0.4480836153107755

    6k_fact = math.factorial(k)7numerator = lam_k * e_neg_lam8prob = numerator / k_fact
    values this step0.4480836153107755numerator

  8. prob ← 0.22404180765538775

    7numerator = lam_k * e_neg_lam8prob = numerator / k_fact9print('RESULT:', round(prob, 6))
    values this step0.22404180765538775prob

  9. stdout ← RESULT: 0.224042

    8prob = numerator / k_fact9print('RESULT:', round(prob, 6))
    values this stepRESULT: 0.224042stdout

With the library

scipy.stats.poisson.pmf(k, mu) takes k first, then the rate parameter (named mu in scipy). Applies the same formula with full floating-point precision.

library.py 
from scipy.stats import poisson
from dalib.display import set_display
set_display()

lam = 3
k = 2
prob = poisson.pmf(k, lam)
print('lambda:', lam, ' k:', k)
print('RESULT:', round(prob, 6))

lambda: 3  k: 2
RESULT: 0.224042

Implementation notes

  • The Poisson distribution models counts of rare, independent events in a fixed interval (e.g. 2 phone calls in a minute given mean rate 3/min). λ is both the mean and the variance of the distribution.
  • Poisson as the limit of Binomial: Binomial(n, λ/n) → Poisson(λ) as n→∞. For large n and small p with fixed λ=np, the Poisson approximation is computationally cheaper.
  • math.exp(-lam) and math.factorial(k) are exact-enough for small k and λ. For very large k, use scipy.stats.poisson.pmf to avoid overflow.
  • Cross-reference: binomial-probability (this chapter) for the discrete distribution this approximates in the large-n limit.