Random Variable Calculator

Example data table

Formulas used

• PDF/PMF: describes probability density (continuous) or mass (discrete).
• CDF: F(x)=P(X ≤ x). For continuous, P(a≤X≤b)=F(b)-F(a).
• Mean: E[X]=Σ xᵢpᵢ (discrete) or E[X]=∫ x f(x) dx (continuous).
• Variance: Var(X)=E[X²]-E[X]², with SD=√Var(X).
• Normal: f(x)=1/(σ√(2π)) · exp(-(x-μ)²/(2σ²)).
• Binomial: P(X=k)=C(n,k)pᵏ(1-p)ⁿ⁻ᵏ.
• Poisson: P(X=k)=λᵏ e⁻ˡᵃᵐᵇᵈᵃ / k!.
• Uniform: f(x)=1/(b-a) for a≤x≤b.
• Exponential: f(x)=λe⁻ˡᵃᵐᵇᵈᵃˣ, F(x)=1-e⁻ˡᵃᵐᵇᵈᵃˣ for x≥0.

How to use this calculator

1. Select a distribution and enter its parameters.
2. Enter a point x to get PDF/PMF and CDF.
3. Set x0 and x1 to compute an interval probability.
4. Use q to find a percentile (inverse CDF estimate).
5. Press Submit to show results above this form.
6. Use CSV or PDF buttons to export your output.

Understand random variables in real datasets

A random variable maps outcomes to numbers. This calculator helps you test distribution choices quickly. Use it for labs, quality checks, and forecasting. The output table shows mean, variance, and tail probabilities.

Compare normal, binomial, and Poisson behavior

Normal models measurement noise and aggregated effects. Binomial models fixed trials with success probability. Poisson models event counts per interval using a rate. The plot reveals shape changes when parameters shift.

Use interval probability for decision thresholds

Interval probability estimates how often values fall between x0 and x1. This supports control limits, service targets, and risk bands. Left and right tails highlight extreme outcomes. Discrete inclusivity adjusts boundary counting for integers.

Interpret mean and variance with practical meaning

Mean summarizes the typical value under the model. Variance measures spread around the mean. Standard deviation is the square root of variance. Higher spread increases tail risk, even when the mean stays constant.

Enter custom probability tables for surveys

Custom discrete input supports scored surveys and graded outcomes. Add x and p pairs for each category. Probabilities are normalized for safety. The tool computes E[X] and Var(X) from the table.

Export results for reports and audits

CSV exports clean rows for spreadsheets and dashboards. PDF export creates a printable summary for sharing. Keep examples beside your assumptions. Re-run scenarios to document sensitivity across parameter ranges.

FAQs

1) What does the CDF value mean?
It is the probability that X is less than or equal to x. For discrete models, x is floored for CDF. For continuous models, it follows the curve smoothly.

2) Why does the tool show PDF for some distributions?
Continuous distributions use a density, not a point probability. Probability comes from areas between bounds. The interval output uses CDF differences to compute that area.

3) How should I choose between binomial and Poisson?
Use binomial for a fixed number of trials. Use Poisson for counts over time or space. Poisson often works when trials are many and p is small.

4) What happens if my custom probabilities do not sum to one?
The calculator normalizes them. Each probability is divided by the total. This keeps the distribution valid. Negative probabilities are rejected to prevent incorrect results.

5) What is the quantile result used for?
It returns a cutoff where the CDF reaches q. Use it for percentiles, safety limits, and service levels. For discrete distributions, it returns the smallest x meeting the target probability.

6) Why can discrete inclusivity change results?
With integers, boundary points can be counted or excluded. Inclusive mode uses ≤ and ≥. Exclusive mode uses < and >. Continuous distributions are unaffected because exact boundary probability is zero.