Inputs & Modes
Results
X ~ Binomial(n=10, p=0.5), the probability mass at k=5 is P(X = 5) = 0.24609375.How it’s calculated
- Model the count as
X ~ Binomial(10, 0.5). - Evaluate
P(X=5) = C(10,5) * 0.5^5 * (1-0.5)^5. - The probability mass is
0.24609375, so about24.6%of samples have exactly five successes.
Visualization
Discrete distributions render as bars with highlighted tails; continuous ones display the PDF curve plus shaded regions or quantile markers.
Static example before you compute: for a binomial model with n=10, p=0.5, and PMF at k=5, read the output as the probability of exactly 5 successes. The chart should show bars for possible success counts and highlight the selected count.
Teacher notes
- The step log shows each transform (beta or gamma mapping, Newton updates) so you can discuss the method live.
- Exact intervals for binomial p and Poisson lambda appear next to point estimates.
- Share links, CSV tables, and LaTeX snippets support homework and worked examples.
FAQ
How are CDF and quantile values computed?
Binomial CDF uses regularized incomplete beta. Poisson and chi-square use regularized gamma. Student t quantiles use beta mapping with Newton or bisection.
What can I do with exact intervals?
Use the Clopper-Pearson interval for p and the chi-square interval for lambda to report uncertainty in labs or assessments.
Which distribution should I start with?
Use binomial for fixed trials with success/failure outcomes, Poisson for event counts over a fixed exposure, Student t for mean checks with estimated variance, and chi-square for variance or goodness-of-fit work.
Why can another calculator give a slightly different tail value?
Tail probabilities depend on whether the endpoint is included, whether one-sided or two-sided probability is requested, and how the numerical routine handles extreme probabilities.
How should I report rounded probability results?
Keep enough digits for the decision you are explaining and report the distribution, parameters, tail direction, and rounding together so the result can be reproduced.
How to use the distribution calculator effectively
Choose the model first
Pick the distribution before entering numbers. Binomial needs a trial count and success probability, Poisson needs a rate or expected count, Student t needs degrees of freedom, and chi-square needs a variance or goodness-of-fit context.
Check the tail definition
Most disagreements between tools come from tail direction, endpoint inclusion, and two-sided conventions. Record whether you used left tail, right tail, between, outside, CDF, survival, or quantile mode.
Compare with intervals
Use exact intervals when you need a defensible uncertainty range for a count, rate, or proportion. Keep the original count and confidence level next to the interval so the calculation is easy to audit.
Common mistakes to avoid
Do not mix event rates with observed counts, use a normal approximation without checking the tail, or compare rounded probabilities without matching the same distribution parameters.
Interpretation guidance
Treat very small probabilities as evidence under the selected model, not as proof by themselves. Explain the model assumptions, the tail you tested, and whether the result changes when parameters move slightly.