# What is the joint probability density function for two continuous distributions of iid random variables of different type?

Let $$X$$,$$Y$$ be independent and identically distributed continuos random variables. $$X$$ is distributed according to a Gaussian distribution with pdf $${f_g}_X(x;\mu,\sigma)$$, while $$Y$$ according to a uniform distribution with pdf $${f_u}_Y(y;a,b)$$ in the interval $$[a,b]$$. What is the joint probability density function?

Since $$X$$,$$Y$$ are continuous and indipendent from each other, is th joint pdf the product of the two, even if the two distributions are different? This way: $$f_{XY}(x,y) = {f_g}_X(x;\mu,\sigma) \cdot {f_u}_Y(y;a,b)$$