Proof of two-dimensional probability density function

Let the probability density of a two-dimensional random variable (X,Y) be f(x,y)=e-x-yx>0,y>0;0,other. Seek proof that x, y are independent of each other.

Solution: first calculate the marginal density functions of x and y separately as follows:

Marginal density function of x: x<0 when the marginal density is 0,x>0 as follows:

Similarly, the marginal density function of y can be obtained: y<0 when the marginal density is 0,y>0 when As follows:

Then by

it can be seen that x, y are independent of each other.

Probability density of a two-dimensional random variable?

(1), From the properties of probability density, ∫(0,1)dy∫(-√y,√y)f(x,y)dx=1. ∴c∫(0,1)ydy∫(-√y,√y)x²dx=1.

∴4c/21=1, c=21/4.

(2), The density function of the marginal distributions of X, and Y are, respectively, fX (x)=∫(x²,1)f(x,y)dy=c∫(x²,1)x²ydy=…=(21/8)x²(1-x^4), -1<x<1, fX(x)=0, and x is other.

fY(y)=∫(-√y,√y)f(x,y)dx=c∫(-√y,√y)x²ydx=…=(7/2)y^(5/2), 0<y<1, fY(y)=0, y is other.

For reference.