If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$.

Is this a way to produce a bivariate distribution $f_{(X,Y)}$?

$f_{(X,Y)} = p(X=x \cap Y=y) = p(x) \times p(y) = f_X(x)\times f_Y(y)$.

Does this hold for the case of integrals as well in the continuous case?


Yes, the joint cumulative distribution function $F_{X,Y}(x,y)$, that is, the probability that $X\le x$ and $Y\le y$, is the product of the individual cdf $F_X(x)$ and $F_Y(y)$.

If $X$ and $Y$ have continuous distributions with density functions $f_X(x)$ and $f_Y(y)$, then the joint density function $f_{X,Y}(x,y)$ is the product of the individual density functions.

The discrete case is as you described it.

  • $\begingroup$ I assume it works for the pmf as well, which is what I was implying vaguely above $\endgroup$
    – AER
    Jul 3 '14 at 0:57
  • $\begingroup$ I have added a couple of lines about the pmf (discrete case) and pdf (continuous case). $\endgroup$ Jul 3 '14 at 1:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.