Calculator Inputs
This page uses one centered content flow, while the calculator inputs shift to three columns on large screens, two on medium, and one on mobile.
Formula Used
The calculator applies the change-of-variable identity for a one-to-one monotone mapping. If Y = g(X), then the transformed density is fY(y) = fX(g-1(y)) × |d/dy g-1(y)|.
It also computes transformed moments exactly for the supported combinations. Affine mappings use E[Y] = aE[X] + b and Var(Y) = a²Var(X). Lognormal-compatible cases preserve closed forms under power, reciprocal, and exponential mappings.
When the transformed output no longer belongs to a named standard family, the tool still reports support, inverse mapping, Jacobian term, and exact moments whenever closed-form expressions exist.
How to Use This Calculator
First, choose the base distribution that represents the original random variable X. Enter the required parameters exactly as labeled on the form.
Next, choose the transformation rule. Some mappings need extra constants, such as a scale, shift, exponent, or power. The labels update automatically so the required fields stay clear.
Press the calculate button. The results appear immediately below the header and above the form. Review the transformed mean, variance, support, inverse map, Jacobian term, and density relation.
Use the export buttons to save the result table as CSV or PDF. The reset button clears the form and returns the page to its default state.
Example Data Table
| Example | Base Setup | Transformation | E[Y] | Var(Y) | Support of Y |
|---|---|---|---|---|---|
| Normal with affine mapping | Normal, μ=10, σ=2 | Y = aX + b | 12 | 9 | (−∞, ∞) |
| Uniform with logarithmic mapping | Uniform, a=2, b=8 | Y = ln(X) | 1.54154 | 0.145861 | [0.693147, 2.079442] |
| Exponential with power mapping | Exponential, λ=2.5 | Y = X^k | 0.32 | 0.512 | [0, ∞) |
| Lognormal with reciprocal mapping | Lognormal, μln=0.3, σln=0.6 | Y = 1 / X | 0.88692 | 0.340869 | (0, ∞) |
FAQs
1. What does this calculator evaluate?
It evaluates how a selected transformation changes a random variable’s support, moments, inverse mapping, Jacobian term, and density relation for supported continuous models.
2. Why does the Jacobian matter?
The Jacobian rescales density after transformation. Without it, the new density would not preserve total probability correctly.
3. Why are some combinations blocked?
Some mappings are non-monotone, cross zero, or lack stable closed-form moments in this implementation. Those cases need separate numerical treatment.
4. Are the displayed moments exact?
Yes. For every supported case, the mean and variance shown here come from exact closed-form expressions rather than simulation.
5. Can it handle non-monotone transforms?
No. This version focuses on one-to-one monotone mappings because they produce a direct inverse and a clean Jacobian-based density formula.
6. What is the support of Y?
Support of Y is the full set of transformed values that can occur after applying the selected rule to every valid value of X.
7. Why is reciprocal sensitive near zero?
Because 1/X grows without bound as X approaches zero. If the support touches or crosses zero, the transform becomes unstable or invalid.
8. When does the transformed family stay standard?
It stays standard in cases like affine-normal, affine-uniform, lognormal power transforms, reciprocal-lognormal, and exponential maps of normal inputs.