Continuous Random Variable Calculator

Calculator Inputs

Choose a distribution, enter its parameters, then calculate density, cumulative probability, interval probability, percentile, and moments.

Formula Used

Normal: f(x)=1/(σ√(2π))e^{-((x-μ)^2)/(2σ²)}

Uniform: f(x)=1/(b-a) for a≤x≤b.

Exponential: f(x)=λe^{-λx} for x≥0.

Triangular: Uses minimum, mode, and maximum with piecewise density.

Interval probability is calculated as P(L≤X≤U)=F(U)-F(L). Survival probability is 1-F(x).

How To Use This Calculator

1. Select the continuous distribution that matches your problem.
2. Enter the target x value for PDF, CDF, and survival probability.
3. Enter lower and upper bounds for interval probability.
4. Enter a percentile probability between 0 and 1.
5. Add distribution parameters such as mean, deviation, rate, or limits.
6. Submit the form and review the results above the form.
7. Use the graph to compare density and cumulative behavior.
8. Download CSV or PDF output for reports and records.

Example Data Table

Understanding Continuous Random Variables

A continuous random variable can take any value inside an interval. Its probabilities come from area, not from single points. The curve is called a probability density function. Total area under it equals one. This calculator helps you study common models without long manual integration.

Why Density Matters

For a continuous model, P(X = x) is zero. Useful questions ask about ranges, such as P(a ≤ X ≤ b). The calculator gives left tail, right tail, interval probability, density, percentile, mean, variance, and standard deviation. These results support exams, quality control, risk analysis, service timing, and measurement studies.

Choosing A Distribution

The normal model works well for balanced data near a central average. The uniform model fits cases where every value in a range is equally likely. The exponential model describes waiting time until an event. The triangular model is useful when minimum, most likely, and maximum values are known.

Reading The Graph

The PDF graph shows where outcomes are dense. Taller areas mean nearby values are more likely over a small range. The CDF graph rises from zero to one. It shows accumulated probability. Use the shaded interval result with the graph to explain practical risk.

Practical Workflow

Start by selecting the distribution that matches your situation. Enter validated parameters. Add the target x value, interval bounds, and percentile level. Then submit the form. Review the result cards first. Next, compare formulas with the displayed values. Finally, export the result for reports.

Interpreting Results Carefully

A probability close to one means the event is very likely under the chosen model. A probability close to zero means it is rare. Percentiles convert probability into a boundary value. Moments summarize the whole distribution. Always check units, assumptions, and parameter estimates before using the answer in decisions.

Common Mistakes To Avoid

Do not mix rate with average waiting time. For exponential models, rate equals one divided by mean time. Do not enter a negative standard deviation. Do not reverse lower and upper interval bounds. For triangular models, the mode must stay between the minimum and maximum. Small input changes can move tail probabilities a lot, especially near extreme percentiles for safety.

FAQs