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10.3 Geometric and negative binomial distributions

3 min readLast Updated on July 19, 2024

The geometric distribution models the number of trials needed for the first success in independent Bernoulli trials. It's crucial for understanding scenarios with repeated attempts until a desired outcome occurs, like job interviews or manufacturing quality control.

The negative binomial distribution extends this concept, focusing on the number of failures before a specific number of successes. This makes it useful for analyzing more complex situations, such as sales calls or system reliability testing.

Geometric Distribution

Properties of geometric distribution

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  • Models number of trials needed for first success in independent Bernoulli trials
    • Bernoulli trials have two possible outcomes (success or failure)
    • Success probability pp is constant across trials
    • Trials are independent
  • Probability mass function (PMF): P(X=k)=(1p)k1pP(X = k) = (1 - p)^{k - 1}p, where kk is number of trials needed for first success
  • Mean or expected value: E(X)=1pE(X) = \frac{1}{p}
  • Variance: Var(X)=1pp2Var(X) = \frac{1 - p}{p^2}
  • Memoryless property: probability of additional trials needed for success is independent of previous failed trials

Geometric distribution probability calculations

  • Calculate probability of first success on kk-th trial using PMF: P(X=k)=(1p)k1pP(X = k) = (1 - p)^{k - 1}p
    • If p=0.3p = 0.3, probability of first success on 4th trial is P(X=4)=(10.3)30.3=0.1029P(X = 4) = (1 - 0.3)^{3} \cdot 0.3 = 0.1029
  • Find probability of first success within certain number of trials by summing probabilities for each trial
    • Probability of first success within 3 trials: P(X3)=P(X=1)+P(X=2)+P(X=3)P(X \leq 3) = P(X = 1) + P(X = 2) + P(X = 3)
  • Calculate expected number of trials needed for first success using mean: E(X)=1pE(X) = \frac{1}{p}
    • If p=0.2p = 0.2, expected number of trials is E(X)=10.2=5E(X) = \frac{1}{0.2} = 5

Negative Binomial Distribution

Negative binomial vs geometric distributions

  • Negative binomial models number of failures before rr-th success in independent Bernoulli trials
    • rr is fixed, positive integer representing required successes
  • Success probability pp is constant across trials
  • Trials are independent
  • Geometric distribution is special case of negative binomial where r=1r = 1

Negative binomial probability calculations

  • PMF: P(X=k)=(k+r1r1)pr(1p)kP(X = k) = \binom{k + r - 1}{r - 1}p^r(1 - p)^k, where kk is number of failures before rr-th success
    • If p=0.4p = 0.4 and we want probability of 3 failures before 2nd success: P(X=3)=(3+2121)0.42(10.4)3=0.1296P(X = 3) = \binom{3 + 2 - 1}{2 - 1}0.4^2(1 - 0.4)^3 = 0.1296
  • Mean or expected value: E(X)=r(1p)pE(X) = \frac{r(1 - p)}{p}
  • Variance: Var(X)=r(1p)p2Var(X) = \frac{r(1 - p)}{p^2}
  • Calculate cumulative probabilities by summing individual probabilities for values less than or equal to target value

Applications of geometric and negative binomial distributions

  • Geometric distribution:
    • Model number of defective items before finding non-defective item in manufacturing
    • Determine number of job interviews needed before receiving offer
    • Analyze number of attempts before successfully completing task (free throw shots in basketball)
  • Negative binomial distribution:
    • Model number of unsuccessful attempts before achieving specified successes (sales calls to close certain number of deals)
    • Analyze failures before system experiences specified successes (password attempts before user successfully logs in certain times)
    • Determine number of inspections needed to find specified number of defective items in quality control
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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