Analyze distributions with guided inputs and cumulative probabilities. Compare left, right, and interval probabilities easily. View graphs, save reports, and verify distribution behavior confidently.
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| Distribution | Parameters | Query | Approximate Result |
|---|---|---|---|
| Normal | μ = 0, σ = 1 | P(X ≤ 1) | 0.841345 |
| Binomial | n = 10, p = 0.50 | P(X ≤ 6) | 0.828125 |
| Poisson | λ = 3 | P(X ≤ 2) | 0.423190 |
| Exponential | λ = 0.40 | P(X ≤ 3) | 0.698806 |
| Uniform | a = 2, b = 8 | P(X ≤ 5) | 0.500000 |
1) Normal CDF
F(x) = 0.5 × [1 + erf((x − μ) / (σ√2))]
Point density:
f(x) = [1 / (σ√(2π))] × e^(-0.5((x−μ)/σ)^2)
2) Binomial CDF
P(X ≤ k) = Σ [C(n, i) × p^i × (1−p)^(n−i)], for i = 0 to k
Point mass:
P(X = k) = C(n, k) × p^k × (1−p)^(n−k)
3) Poisson CDF
P(X ≤ k) = Σ [e^(−λ) × λ^i / i!], for i = 0 to k
Point mass:
P(X = k) = e^(−λ) × λ^k / k!
4) Exponential CDF
F(x) = 1 − e^(−λx), for x ≥ 0
Density:
f(x) = λe^(−λx), for x ≥ 0
5) Uniform CDF
F(x) = 0 for x < a
F(x) = (x − a) / (b − a) for a ≤ x ≤ b
F(x) = 1 for x > b
Density:
f(x) = 1 / (b − a) for a ≤ x ≤ b
Interval Probability
Continuous models use P(a ≤ X ≤ b) = F(b) − F(a).
Discrete models use inclusive bounds:
P(a ≤ X ≤ b) = F(b) − F(a−1).
A cumulative distribution gives the probability that a random variable is less than or equal to a chosen value. It is useful for thresholds, percentiles, tail checks, and interval comparisons.
The CDF accumulates probability up to a point. A PDF is a density for continuous variables, while a PMF gives exact point probabilities for discrete variables like binomial or Poisson counts.
Use interval mode when you need the probability between two bounds, such as exam scores between 60 and 80, or defect counts from 2 through 5.
Those distributions model count data. Counts happen in whole numbers, so the calculator rounds down to valid integer boundaries and treats interval endpoints as inclusive.
Continuous distributions, such as normal and exponential, produce smooth cumulative curves. Discrete distributions, such as binomial and Poisson, jump at integer values because probability accumulates in steps.
Yes. Choose the right-tail option to compute the probability at or above the selected threshold. This is useful in reliability, quality control, and statistical hypothesis work.
After calculation, you can download a CSV summary for spreadsheets or a PDF report for sharing, printing, recordkeeping, and documentation.
Keep probabilities between 0 and 1, standard deviation greater than 0, rates positive, and uniform minimum smaller than maximum. For intervals, use a lower bound that does not exceed the upper bound.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.