Estimation of Lower Bound in Poisson Distribution

Hypergeometric Explorer

Sampling without replacement from a finite population.

Population Size (N)

Successes in Pop (K)

Sample Size (n)

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Find Probability: Exact (P = x) Cumulative (P ≤ x) Survival (P ≥ x)

Observed Successes (x)

Probability: 0.0000

Theory: Hypergeometric Distribution

Topic: Discrete Distributions | Difficulty: Intermediate

The Hypergeometric distribution models the number of successes ($k$) in a sample of size $n$ drawn without replacement from a population of size $N$ containing $K$ successes. Unlike the Binomial distribution, the probability of success changes with each draw.

Probability Mass Function (PMF)

` P(X=k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}`

Where $\binom{n}{k}$ is the binomial coefficient.
Support: $\max(0, n+K-N) \le k \le \min(n, K)$

Moments

• Mean: $n \frac{K}{N}$
• Variance: $n \frac{K}{N} \frac{N-K}{N} \frac{N-n}{N-1}$
• Mode: $\lfloor \frac{(n+1)(K+1)}{N+2} \rfloor$

Shape Parameters

• Skewness: $\frac{(N-2K)(N-1)^{1/2}(N-2n)}{[nK(N-K)(N-n)]^{1/2}(N-2)}$
• Excess Kurtosis: Complex function involving $N, n, K$. Generally closer to 0 as $N \to \infty$.

Comparison with Binomial

Finite Population Correction: The key difference in variance is the term $\frac{N-n}{N-1}$.

1. If $N$ is very large compared to $n$ (sample is < 5% of population), this term $\approx 1$.
2. In this case, the Hypergeometric distribution converges to the Binomial Distribution $B(n, p=K/N)$.

Python Implementation

Python (Scipy)

import scipy.stats as stats

N = 50   # Population Size
K = 10   # Total Successes in Pop
n = 5    # Sample Size
x = 2    # Observed Successes

# 1. Exact Probability P(X=x) (PMF)
prob = stats.hypergeom.pmf(x, N, K, n)
print(f"P(X={x}) = {prob:.4f}")

# 2. Cumulative Probability P(X<=x) (CDF)
cum_prob = stats.hypergeom.cdf(x, N, K, n)
print(f"P(X<={x}) = {cum_prob:.4f}")