Transformation of random variables

[matiasmorant]

we know that, if, for example, the variable X has a probability distribution f and that the variable Y has a probability distribution g, and both are independent then the variable Z=X+Y has a distribution f*g, where " * " stands for convolution. if Z=XY then the probability distribution of Z is $$\int_{-\infty }^{\infty } \frac{f\left[\frac{z}{x}\right]g[x]}{|x|} \, dx$$ well... in any case, if we have that Z=f[X,Y], then we can obtain the probability dist. of Z as a function of the prob.dist. of X and Y

now, the question is: let A be the event Z<z; B, the event Y<y; C, the event X<x;

and $$P(A)=P(B\cup C)=P(B)+P(C)-P(B)P(C)$$

what is the relationship between variables Z, X and Y ? that is, what is " f " in Z=f[X,Y] ?

 

[Valkarie]

The relationship between variables Z, X and Y is that Z can be expressed as a function of X and Y, i.e., Z = f(X,Y). The exact form of the function will depend on how Z is related to X and Y. For example, if Z = X + Y, then f(X,Y) = X + Y; if Z = XY, then f(X,Y) = XY.