Say I have two independent random variables $X$ and $Y$ both having the exponential distribution. I.e.
$f_X(x) = \lambda_1 e^{-\lambda_1 x}, \ x \ge 0, 0$ elsewhere
$f_Y(y) = \lambda_2 e^{-\lambda_2 y}, \ x \ge 0, 0$ elsewhere
Does this mean that the joint distribution is the following
$f_{X, Y}(x, y) = \lambda_1 \lambda_2 e^{-\lambda_1 x - \lambda_2 y}, \ x, y \ge 0, 0$ elsewhere
I.e. you can simply multiply the functions?