StatSolver
Enter values for P(X,Y).
In my probability course I encountered a difficult problem that required looking at two variables at the same time. I was analyzing a software program that had two different modules. I needed to track the number of errors in Module 1 (variable X) versus the number of errors in Module 2 (variable Y). That assignment taught me that looking at one variable in isolation often hides the real story. I built this calculator to handle those complex "X and Y" scenarios where you need to check if two events are truly independent or if they influence each other. For example I had to calculate if a bug in the first module made it more likely to find a bug in the second module.
Joint distribution guide
A joint probability distribution describes two discrete random variables at the same time. Each cell is the probability of one specific pair of outcomes, written as P(X = x, Y = y). The calculator turns that table into marginal distributions, expected values, covariance, correlation, conditional probabilities, and an independence check.
If the total is below 1, an outcome is missing. If it is above 1, the entries cannot describe a valid probability model. StatSolver checks this before calculating any summary.
A marginal distribution collapses one dimension. Add across a row to obtain P(X = x), or down a column to obtain P(Y = y). A conditional probability narrows the model after an outcome is known: P(Y = y | X = x) = P(X = x, Y = y) / P(X = x).
X and Y are independent only when P(X = x, Y = y) = P(X = x)P(Y = y) for every pair—not merely for one convenient cell. A single mismatch is enough to establish dependence.
Suppose the four probabilities for (X,Y) = (0,0), (0,1), (1,0), and (1,1) are 0.4, 0.1, 0.2, and 0.3. They sum to 1. The marginals are P(X = 1) = 0.5 and P(Y = 1) = 0.4.
Independence would require P(X = 1, Y = 1) = 0.5 × 0.4 = 0.2, but the table gives 0.3. The variables are therefore dependent. Here E[XY] = 0.3, E[X]E[Y] = 0.2, so the covariance is 0.1 and the correlation is approximately 0.4082.
Covariance indicates the direction of linear co-movement, while correlation rescales that relationship to the interval from -1 to 1. Neither statistic proves causation. A covariance of zero also does not generally prove independence, which is why StatSolver checks the full probability table separately.
The distribution of Z = X + Y combines cells that produce the same sum. Use it when the total outcome matters, such as total defects across two modules or total demand across two locations.
Do not enter percentages as whole numbers: use 0.25 for 25%. Do not use a correlation result by itself as an independence test. When a conditional denominator is zero, that conditional probability is undefined. For a single yes/no event, use the Bernoulli calculator; for a fixed number of repeated trials, use the Binomial calculator.