This calculator helps you compute the marginal distribution for a discrete joint probability distribution.
Ensure the number of probabilities matches the product of the number of X and Y values.
Understanding Marginal Distributions
In probability theory and statistics, a marginal distribution describes the probability distribution of a single random variable from a set of random variables, ignoring the information about the other variables. When we have a joint probability distribution that describes the likelihood of two or more random variables occurring together, the marginal distribution tells us the probability of any given value for one of those variables on its own.
What is a Joint Probability Distribution?
A joint probability distribution, P(X, Y), gives the probability that random variable X takes on a specific value x and random variable Y takes on a specific value y simultaneously. For discrete random variables, this is often represented in a table (a joint probability table).
How to Calculate Marginal Distributions
To find the marginal distribution of a single variable (e.g., X), you sum the joint probabilities over all possible values of the other variable(s) (e.g., Y).
The marginal probability of X = x, denoted as P(X=x), is calculated by summing P(X=x, Y=y) for all possible values of y:
P(X=x) = Σ P(X=x, Y=y) for all y
Similarly, the marginal probability of Y = y, denoted as P(Y=y), is calculated by summing P(X=x, Y=y) for all possible X values:
P(Y=y) = Σ P(X=x, Y=y) for all x
Calculator Logic
Our calculator takes the distinct values for your two random variables (X and Y) and their corresponding joint probabilities. It then applies the summation rules described above to compute and display the marginal probabilities for each variable.
The joint probabilities are expected to be entered in a flattened, row-by-row order. For example, if X has values {x1, x2} and Y has values {y1, y2}, the order of joint probabilities should be P(x1, y1), P(x1, y2), P(x2, y1), P(x2, y2).
Use Cases
Marginal distributions are fundamental in understanding complex systems. They are used in:
Risk Management: Analyzing the probability of individual risk factors occurring independently.
Machine Learning: Feature selection and understanding the individual behavior of variables.
Econometrics: Studying the impact of single economic indicators.
Experimental Design: Isolating the effects of one variable when multiple factors are involved.
Example Calculation:
Let's say we have two random variables:
Variable X representing the number of defective items in a sample (values: 0, 1)
Variable Y representing the type of defect (values: A, B)
';
// Check if marginal sums are close to 1
var sumMarginalX = Object.values(marginalX).reduce(function(sum, prob) { return sum + prob; }, 0);
var sumMarginalY = Object.values(marginalY).reduce(function(sum, prob) { return sum + prob; }, 0);
if (Math.abs(sumMarginalX – 1.0) > 1e-9) {
resultHtml += 'Warning: Marginal probabilities for X sum to ' + sumMarginalX.toFixed(4) + '.';
}
if (Math.abs(sumMarginalY – 1.0) > 1e-9) {
resultHtml += 'Warning: Marginal probabilities for Y sum to ' + sumMarginalY.toFixed(4) + '.';
}
resultDiv.innerHTML = resultHtml;
}