Analysis of Joint Probability Density Function and Marginal PDFs

What does it mean when the two marginal pdfs are each normal?

Can x and y be normal individually but their joint pdf is not bivariate normal?

Answer:

When the two marginal pdfs are each normal, it indicates that the individual random variables x and y follow a normal distribution when considered separately. However, the joint pdf of x and y may not necessarily imply a bivariate normal distribution.

A joint probability density function (pdf) describes the probabilities of two or more random variables occurring together. To show that the function f(x, y) is a joint pdf, we need to verify that it satisfies two conditions: f(x, y) is non-negative for all x and y, and the integral of f(x, y) over its entire domain is equal to 1.

If the two marginal pdfs, f(x) and f(y), are each normal, it means that the individual random variables x and y follow a normal distribution when considered separately. However, the joint pdf of f(x, y) does not necessarily imply a bivariate normal distribution.