Switch between discrete and continuous joint models easily. See conditional probabilities and moments in seconds. Download clean reports for classwork, audits, and decisions today.
| xi \ yj | 0 | 1 | 2 |
|---|---|---|---|
| 0 | 0.05 | 0.10 | 0.05 |
| 1 | 0.10 | 0.40 | 0.10 |
| 2 | 0.05 | 0.10 | 0.05 |
Bivariate distributions describe how two variables move together. In a 3×3 joint table that sums to 1.000000, marginals compress the table into P(X) and P(Y). This calculator reports E[X], E[Y], Var(X), Var(Y), Cov(X,Y), and Corr(X,Y) for quick interpretation of dependence.
Enter probabilities P(X=xi,Y=yj) as a matrix. The tool computes P(X=xi)=ΣjPij and P(Y=yj)=ΣiPij. For the sample table, the center cell 0.40 dominates mass, producing higher marginal probabilities at xi=1 and yj=1. Normalization can convert raw counts into valid probabilities.
Conditionals answer “given Y, what is X?” The calculator returns P(X=xi|Y=yj)=Pij/P(Y=yj) and P(Y=yj|X=xi)=Pij/P(X=xi). Choose row and column indices to see how probability shifts. If P(Y=yj) is small, conditional estimates become unstable, so interpret carefully.
Expected values use weighted sums: E[XY]=ΣiΣj xi·yj·Pij and Cov=E[XY]−E[X]E[Y]. Correlation standardizes covariance by √(VarX·VarY), making values comparable across scales. A near‑zero correlation does not guarantee independence, but it flags weak linear association.
For continuous modeling, the bivariate normal is defined by μx, μy, σx, σy, and ρ. The calculator evaluates the joint density f(x,y) at a chosen point and reports conditional normals: X|Y=y has mean μx+ρ(σx/σy)(y−μy) and variance σx²(1−ρ²). Similar formulas apply for Y|X=x.
Exact bivariate CDF regions can be complex. The rectangle option estimates P(xmin≤X≤xmax, ymin≤Y≤ymax) using Monte Carlo sampling with n between 2,000 and 200,000. The tool reports an estimated probability p and standard error √(p(1−p)/n), helping you judge precision and increase n when needed. Reliable estimates guide grading, QA checks, and reporting across teams. Use consistent units, document assumptions, and save exports for review. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits and collaboration. Repeatability improves transparency for audits.
1) What inputs are required for discrete mode?
Provide a joint probability matrix. Optionally add X and Y support values. The tool can normalize sums that differ from 1, then computes marginals, moments, and conditionals.
2) Why does the matrix need to sum to one?
A joint probability table represents a full distribution, so total mass must equal 1. If you entered counts, enable normalization to convert them into probabilities.
3) How are covariance and correlation interpreted?
Covariance measures joint variation in original units. Correlation rescales it to −1 to 1, showing linear association strength. Zero correlation can still occur with non‑linear dependence.
4) What does “conditional distribution” mean here?
It is the distribution of one variable after fixing the other. For example, P(X|Y=yj) divides the joint column by P(Y=yj), producing probabilities that sum to 1.
5) What does the normal mode output represent?
It computes the bivariate normal density at a chosen point and the conditional normal parameters. If rectangle estimation is enabled, it approximates region probability by random sampling.
6) How accurate is the Monte Carlo rectangle estimate?
Accuracy improves with larger n. The standard error √(p(1−p)/n) is reported, so you can increase samples until the error is acceptable for your use case.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.