Copula Model Preview

Math Probability • Non Parametric and Computational Probability

Written by STEM Calculators Team Published February 15, 2026 Updated February 24, 2026

Preview dependence with a copula by simulating a Gaussian copula for two uniform marginals \(U,V\in[0,1]\). Estimate a joint probability (like \(P(U>u_0,\;V>v_0)\)) and visualize it on an animated dependence scatter plot.

The plot shows the threshold lines \(u_0\) and \(v_0\) and displays the computed probability inside the graph box.

Copula type

Gaussian copula: correlate standard normals, then transform with \(\Phi\) to get correlated uniforms.

Correlation \(\rho\) (Gaussian copula)

Use \(-0.99\le\rho\le0.99\). For independence, \(\rho\) is ignored.

Event to compute

Thresholds \(u_0,v_0\) are shown as vertical/horizontal lines on the scatter plot.

Threshold \(u_0\) (0 to 1)

Threshold \(v_0\) (0 to 1)

Simulation trials \(N\)

Larger \(N\) reduces Monte Carlo error (roughly \(\propto 1/\sqrt{N}\)). Limit: 200,000.

Repeatable randomness Use deterministic seed

If checked, repeated runs with the same inputs produce the same simulated estimate.

Ready

Dependence scatter plot (animated) + joint probability

Play speed Animation ready

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The badge inside the plot reports the analytic copula probability (approx) and the simulated estimate (with standard error). The vertical and horizontal lines show \(u_0\) and \(v_0\).

Enter inputs and click “Calculate”.

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Copula model preview (bivariate dependence)

Copulas in one sentence

A copula is a function that couples marginal distributions into a joint distribution by describing dependence separately from the marginals.

\[ \text{Copula:}\quad C(u,v)=P(U\le u,\;V\le v),\quad U,V\sim \text{Uniform}(0,1). \]

The calculator focuses on \(U,V\in[0,1]\) directly so you can see the dependence geometry.

Sklar’s theorem (why copulas matter)

Sklar’s theorem says that for any joint CDF \(H(x,y)\) with marginal CDFs \(F_X(x)\) and \(F_Y(y)\), there exists a copula \(C\) such that:

\[ H(x,y)=C\big(F_X(x),\;F_Y(y)\big). \]

If the marginals are continuous, the copula \(C\) is unique. This separation is valuable in applications like finance and risk management: you can keep your chosen marginals but swap dependence models.

Transform view. Define

\[ U=F_X(X),\qquad V=F_Y(Y). \]

Then \(U,V\) are uniform on \([0,1]\), and their joint CDF is exactly the copula: \(C(u,v)=P(U\le u,\;V\le v)\).

Gaussian copula (the one used here)

The Gaussian copula is built from a correlated bivariate normal distribution. The recipe is:

Step 1. Correlated normals

\[ \begin{aligned} Z_1 &\sim N(0,1),\\ Z_2 &= \rho Z_1+\sqrt{1-\rho^2}\,\varepsilon,\qquad \varepsilon\sim N(0,1)\ \text{independent}. \end{aligned} \]

Step 2. Transform to uniforms

\[ U=\Phi(Z_1),\qquad V=\Phi(Z_2), \]

where \(\Phi\) is the standard normal CDF. The pair \((U,V)\) has uniform marginals but retains dependence controlled by \(\rho\).

Copula formula. The Gaussian copula is

\[ C(u,v)=\Phi_{\rho}\!\left(\Phi^{-1}(u),\;\Phi^{-1}(v)\right), \]

where \(\Phi_\rho\) is the bivariate normal CDF with correlation \(\rho\). In the calculator, this is approximated numerically via a stable 1D integral representation.

From \(C(u_0,v_0)\) to the event probability you want

Many useful joint probabilities can be expressed using \(C(u_0,v_0)\) plus complements. For example, since \(P(U\le u_0)=u_0\) and \(P(V\le v_0)=v_0\) (uniform marginals), we have:

\[ \begin{aligned} P(U\le u_0,\;V\le v_0) &= C(u_0,v_0),\\ P(U>u_0,\;V>v_0) &= 1-u_0-v_0+C(u_0,v_0),\\ P(U\le u_0,\;V>v_0) &= u_0-C(u_0,v_0),\\ P(U>u_0,\;V\le v_0) &= v_0-C(u_0,v_0). \end{aligned} \]

The scatter plot shows the threshold lines \(u_0\) and \(v_0\). Points falling in the selected region are highlighted, and the computed probability is displayed inside the plot.

Monte Carlo simulation (why the estimate converges)

Simulation estimates the probability by repeated sampling. If \(I_i\) is the indicator that the \(i\)-th sample satisfies the event, then

\[ \hat p=\frac{1}{N}\sum_{i=1}^{N} I_i. \]

By the law of large numbers, \(\hat p\to p\) as \(N\to\infty\). A rough uncertainty scale is given by the Monte Carlo standard error:

\[ \mathrm{SE}\approx \sqrt{\frac{\hat p(1-\hat p)}{N}}. \]

This is why increasing \(N\) improves stability. The calculator reports \(\mathrm{SE}\) and a simple \(\hat p\pm 1.96\,\mathrm{SE}\) band.

How to use this preview with real marginals

In practice, you often want \(X\) and \(Y\) with non-uniform marginals \(F_X\) and \(F_Y\). Once you have \(U,V\) from a copula, you can map back to your desired marginals using inverse CDFs:

\[ X=F_X^{-1}(U),\qquad Y=F_Y^{-1}(V). \]

This calculator keeps marginals as uniforms so the dependence is easy to see on the \([0,1]\times[0,1]\) plot. In advanced settings (university level), this idea extends to higher dimensions and to constructions like vine copulas.

Dependence scatter plot (animated) + joint probability

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