Question

# Let X and Y be jointly continuous random variables wth joint PDF is given by: f_{X,Y}(x,y)=2 where 0 \leq y \leq x \leq 1 Show the marginal PDF of X and Y.

Random variables
Let X and Y be jointly continuous random variables wth joint PDF is given by:
$$f_{X,Y}(x,y)=2$$ where $$0 \leq y \leq x \leq 1$$
Show the marginal PDF of X and Y.

2021-05-02

It is given that X and Y be jointly continuous random variables with joint PDF is given by:
$$f_{X,Y}(x,y)=2\ where\ 0 \leq y \leq x \leq 1$$
We have to find marginal PDF of X and Y.
$$f_{X}(x)=\int_{a}^{b}f(x,y)dy\ and\ f_{Y}(y)=\int_{a}^{b}f(x,y)dx$$
$$f_{X}(x)=\int_{0}^{1}2dy\ and\ f_{Y}(y)=\int_{0}^{1}2dx$$
$$f_{X}(x)=2\ and\ f_{Y}(y)=2$$
Thus, the marginal PDF of X and Y are:
$$f_{X}(x)=\begin{cases}2,\ 0\leq x\leq 1\\0,\ otherwise\end{cases}$$
$$f_{Y}(y)=\begin{cases}2,\ 0\leq y\leq 1\\0,\ otherwise\end{cases}$$