Let $X,Y$ are independent uniform random variables on $[0,1]$.
How do I find firstly the joint density function then secondly the joint distribution function.
I know $$f_X(x)=\begin{cases}1&:&x\in[0,1]\\0 &:&\text{otherwise}\end{cases}$$
$$f_Y(y)=\begin{cases}1&:&y\in[0,1]\\0 &:&\text{otherwise}\end{cases}$$
but I am unsure how to get the joint density function firstly then I am not 100% sure how to get the distribution (although I suspect you integrate the density function).
Best Answer
By independence, the joint density function is $f_{X,Y}(x,y) = f_X(x) f_Y(y)$. Be sure to incorporate the information about the support of $f_X$ and $f_Y$ (where it is zero and where it is nonzero).
Similarly, the joint distribution function is $P(X\le x, Y \le y) = P(X \le x) P(Y \le y)$ by independence. Again, be careful: $P(X\le x)$ is piecewise linear.