Calculator Input
Enter X values, Y values, and a matching matrix. Use probabilities directly or choose frequency mode.
Example Data Table
This example uses defect count as X and severity score as Y.
| X \ Y | 10 | 20 | 30 |
|---|---|---|---|
| 0 | 0.10 | 0.08 | 0.07 |
| 1 | 0.12 | 0.18 | 0.10 |
| 2 | 0.09 | 0.11 | 0.15 |
The values sum to 1. This makes the table a valid joint distribution.
Formula Used
For a discrete joint distribution, the expected value of XY is the weighted average of all XY products. Each weight is the joint probability for that pair.
Related supporting formulas are also useful during analysis.
In process studies, these values reveal how two quality variables move together.
How to Use This Calculator
Enter the possible X values in the first field.
Enter the possible Y values in the second field.
Paste the matrix with one row for each X value.
Choose probabilities or frequencies based on your data source.
Enable normalization when small rounding issues exist.
Press calculate to view E(XY), marginals, and covariance.
Use the chart to inspect high-probability cells quickly.
Download CSV or PDF for reporting and audits.
Frequently Asked Questions
1. What does E(XY) represent?
E(XY) is the expected value of the product of X and Y. It measures their average combined effect across all possible paired outcomes in the joint distribution.
2. Can I use frequencies instead of probabilities?
Yes. Choose frequency mode and enter raw counts. The calculator converts the counts into joint probabilities before computing E(XY) and related metrics.
3. Why must the probability matrix sum to 1?
A valid joint probability distribution assigns all probability mass across every possible pair. That total must equal 1, otherwise the expectation is not mathematically valid.
4. What happens if rounding causes a slight total mismatch?
Turn on normalization. The calculator rescales every cell by the total matrix sum. This helps when your source table is rounded but still conceptually valid.
5. How is this useful in quality control?
It helps evaluate paired process variables. Examples include defect count with severity score, inspection time with failure grade, or batch size with rejected units.
6. Does a higher E(XY) always mean a bad process?
Not always. Interpretation depends on what X and Y represent. Higher values may indicate more risk, more productivity, or simply larger operating levels.
7. What is the difference between E(XY) and E(X)E(Y)?
E(XY) uses the full joint distribution. E(X)E(Y) multiplies separate averages. Their difference forms covariance and helps show dependence between variables.
8. Can this calculator test independence?
It does not perform a full independence test. However, covariance and correlation provide quick clues about whether X and Y may move together.