List of basic formulas related to the hypergeometric probability distribution:

  1. 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.
  1. Mathematical expectation:
    [
    E(X) = n \cdot \frac{K}{N}
    ]
  2. Variance:
    [
    Var(X) = n \cdot \frac{K}{N} \cdot \left(1 – \frac{K}{N}\right) \cdot \frac{N – n}{N – 1 }
    ]
  3. 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)
    ]

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