List of basic formulas related to the hypergeometric probability distribution:
- The probability formula for the hypergeometric distribution is:
[
P(X = k) = \frac{{\binom{K}{k} \cdot \binom{NK}{nk}}}{{\binom{N}{n}}}
]
where:
- ( N ) — total number of objects,
- ( K ) — the number of successful objects (for example, “white” balls),
- ( n ) — the number of objects selected for sampling,
- ( k ) — the number of successful objects in the sample.
- Mathematical expectation:
[
E(X) = n \cdot \frac{K}{N}
] - Variance:
[
Var(X) = n \cdot \frac{K}{N} \cdot \left(1 – \frac{K}{N}\right) \cdot \frac{N – n}{N – 1 }
] - Cumulative Distribution Function:
To find the probability that a random variable ( X ) will take a value less than or equal to ( k ):
[
P(X \leq k) = \sum_{i=0}^{k} P(X = i)
]