Enter Distribution Inputs
Use the responsive form below. Large screens show three columns, medium screens show two, and mobile screens stack one field per row.
Example Data Table
| Distribution | Parameters | x | Interval | p | PDF at x | CDF at x | Interval Probability | Quantile xp |
|---|---|---|---|---|---|---|---|---|
| Normal | μ = 50, σ = 8 | 54 | [45, 60] | 0.90 | 0.04400817 | 0.69146246 | 0.62836470 | 60.252412 |
| Exponential | λ = 0.40 | 2.5 | [1, 4] | 0.80 | 0.14715178 | 0.63212056 | 0.46842353 | 4.023595 |
| Uniform | a = 10, b = 18 | 13 | [11, 16] | 0.75 | 0.12500000 | 0.37500000 | 0.62500000 | 16.000000 |
Formula Used
General outputs: density f(x), cumulative probability F(x), interval probability P(L ≤ X ≤ U) = F(U) - F(L), and quantile xp where F(xp) = p.
Normal Distribution
f(x) = [1 / (σ√(2π))] e^(-((x-μ)^2 / 2σ^2))
F(x) = 0.5[1 + erf((x-μ)/(σ√2))]
E[X] = μ, Var(X) = σ², xp = μ + σΦ⁻¹(p)
Exponential Distribution
f(x) = λe^(-λx) for x ≥ 0
F(x) = 1 - e^(-λx) for x ≥ 0
E[X] = 1/λ, Var(X) = 1/λ², xp = -ln(1-p)/λ
Uniform Distribution
f(x) = 1/(b-a) for a ≤ x ≤ b
F(x) = (x-a)/(b-a) for a ≤ x ≤ b
E[X] = (a+b)/2, Var(X) = (b-a)²/12, xp = a + p(b-a)
How to Use This Calculator
- Choose Normal, Exponential, or Uniform from the distribution menu.
- Enter the matching distribution parameters, such as mean and standard deviation.
- Provide an x value to evaluate the density, cumulative probability, and survival function.
- Enter the lower and upper interval bounds to measure probability across a range.
- Set the quantile probability
pbetween 0 and 1 to find the corresponding cutoff value. - Click the calculate button to place the result summary above the form.
- Review the table, interpretation, and chart, then export the result as CSV or PDF.
FAQs
1. What does the density value mean?
Density shows how concentrated probability is near x. For continuous models, density is not the probability at one exact point.
2. Why is interval probability more useful than point probability?
A continuous variable has zero probability at any exact point. Useful probability comes from areas across intervals, such as between two bounds.
3. When should I use a normal model?
Use it for symmetric data around a mean, especially when values cluster centrally and extreme values become less likely smoothly.
4. When is the exponential model appropriate?
It works well for waiting times, lifetimes, or arrival gaps when the process is memoryless and the event rate stays constant.
5. What is the main use of a uniform model?
Use it when every value inside a fixed range is equally likely and values outside the range are impossible.
6. What does the quantile output represent?
The quantile xp is the cutoff where cumulative probability reaches p. For example, p = 0.90 gives the 90th percentile.
7. Why might the coefficient of variation be undefined?
It divides the standard deviation by the mean magnitude. When the mean is zero, that ratio becomes undefined or unstable.
8. Does the chart help verify my interval choice?
Yes. The shaded region highlights the chosen interval under the density curve, making it easier to judge how much probability it contains.