It depends on whether x and y are independent of each other. If they are not independent, it is a double integral. The integrand function is the product of the probability density function of these two functions, and then multiplied by xy. If they are independent, this double integral is equivalent to the product of the edge distribution functions of these two functions.

If a two-dimensional random variable (x, y) is regarded as the coordinates of a random point on a plane, then the function value of the distribution function F(x, y) at (x, y) is the probability that the random point (x, y) falls in an infinite rectangular domain with its vertex at the lower left of the point.

In probability theory, the joint distribution of two random variables X and Y is the probability distribution of X and Y..

Extended data:

The normal curve is bell-shaped, with low ends and high middle, which is symmetrical left and right, so people often call it bell-shaped curve.

If the random variable X obeys the normal distribution with a mathematical expectation of μ and a variance of σ 2, it is recorded as N(μ, σ 2). The expected value μ of probability density function with normal distribution determines its position, and its standard deviation σ determines its distribution amplitude. When μ = 0 and σ = 1, the normal distribution is standard normal distribution.

C.F. Gauss derived it from another angle when studying the measurement error, and P.S. Laplace and Gauss studied its properties. It is a very important probability distribution in mathematics, physics, engineering and other fields, and has great influence in many aspects of statistics.