Parameter Inputs
n ≥ 0 • 0 ≤ p ≤ 1 • k ∈ {0, …, n}
Enter as decimal between 0 and 1.
Evaluates P(lower ≤ X ≤ upper) when both bounds valid.
Summary & Moments
- Expected value E[X]
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- Variance Var[X]
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- Standard deviation
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- Mode (ties possible)
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Probability Outputs
- PMF P(X = k)
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- CDF P(X ≤ k)
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- P(lower ≤ X ≤ upper)
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- Upper tail P(X ≥ k0)
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- Lower tail P(X ≤ k0)
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- Quantile k for target p
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Random Sample Generator
Simulates draws using inverse transform; limited to 1000 samples.
How to Use
- Provide the number of Bernoulli trials n and success probability p. For fair processes, p = 0.5.
- Enter the outcome count k to evaluate exact PMF and CDF values instantly.
- Use the bounds or tail sections to examine ranges such as “at least k successes” or “between a and b successes”.
- Supply a probability 0 < p < 1 to find the minimum k where P(X ≤ k) ≥ p (inverse CDF/quantile).
- Generate random samples to approximate the distribution empirically; results appear comma-separated for quick copy/paste.
Formula References
- Ross, S. M. (2014). A First Course in Probability (9th ed.). Pearson.
- Johnson, N. L., Kotz, S., & Kemp, A. W. (1992). Univariate Discrete Distributions (2nd ed.). Wiley.