Advanced Conditional Density Calculator

Calculator Inputs

x value

y value

Correlation ρ

Mean of X

Mean of Y

Standard deviation of X

Standard deviation of Y

Conditional Density Graph

Example Data Table

x	y	Mean X	Mean Y	SD X	SD Y	ρ	Purpose
1.20	0.80	0.00	0.00	1.50	1.10	0.45	Baseline classroom example
0.40	1.10	0.00	0.50	1.20	0.90	0.30	Moderate positive association
-0.70	0.20	-0.20	0.00	0.95	1.30	-0.40	Negative correlation scenario

Formula Used

For continuous variables, conditional density is: f_X|Y(x|y) = f_X,Y(x,y) / f_Y(y), provided that f_Y(y) > 0.

This page models the pair (X,Y) with a bivariate normal distribution. Under that model:

E[X|Y=y] = μ_X + ρ(σ_X/σ_Y)(y - μ_Y)

Var[X|Y=y] = σ_X²(1 - ρ²)

The calculator first evaluates the joint density and marginal density, then verifies the same result using the normal conditional density built from the conditional mean and conditional variance.

How to Use This Calculator

Enter the point x where you want the conditional density.
Enter the observed value y used for conditioning.
Provide means and standard deviations for both variables.
Enter a correlation between -1 and 1, excluding endpoints.
Click the calculate button to show results above the form.
Review conditional density, joint density, marginal density, and derived moments.
Use the graph to inspect the shape of the conditional distribution.
Export the displayed results as CSV or PDF when needed.

Frequently Asked Questions

1. What does conditional density mean?

It describes how likely nearby values of X are when Y is fixed at a specific value. It is a density, not a direct probability.

2. Why is this useful in maths?

It helps analyze dependent continuous variables, update distributions after observations, and study regression-style relationships in probability theory and statistics.

3. Does the result equal a probability?

No. A density can be greater than one. Probabilities come from integrating the density across an interval.

4. Why do I need a marginal density?

The marginal density of Y normalizes the joint density. Without it, you cannot form the conditional density ratio correctly.

5. What restrictions apply to correlation?

The entered correlation must stay strictly between minus one and one. The endpoints make the covariance structure degenerate.

6. Why are means and standard deviations required?

They define the bivariate normal model. Changing them shifts the center and spread of both the joint and conditional distributions.

7. What does the graph show?

It plots the conditional density of X after Y is fixed. The peak occurs near the conditional mean, and width reflects conditional uncertainty.

8. When should I use exported results?

Use them for class notes, audits, reports, model documentation, or quick comparisons between several input scenarios.