# How to find the marginal PDF of $f_X(x)$ in this case?

The random variables X and Y are described by a joint PDF which is uniform on the triangular set defined by the constraints 0 ≤ x ≤ 1, 0 ≤ y ≤ x. How can I find the marginal PDF of $$f_X(x)$$ when the formula for the joint PDF is not given? The PDF is uniform across the stated region, meaning its value is a constant there. The joint PDF is thus:

$$f_{X,Y}(x,y)=\begin{cases} c,& \mbox{if } 0\leq y\leq x\textrm{ and }0\leq x\leq 1,\\ 0,& \mbox{otherwise,}\end{cases}$$

Solving for $$c$$:

$$\displaystyle \int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty f_{X,Y}(x,y)\,dy\,dx = c\int\limits_x \int\limits_{y} dy\,dx = c\int\limits_0^1 \int\limits_0^x \,dy\,dx = c\left.\int\limits_0^1 y\right|_0^x \,dx = c\int\limits_0^1 x\,dx$$

$$= \left.c\cdot\frac{x^2}{2}\right|_0^1 = \frac{c}{2} = 1\Rightarrow c = 2$$

Thus, $$f_{X,Y}(x,y)=\begin{cases} 2,& \text{if } 0\leq y\leq x\textrm{ and }0\leq x\leq 1,\\ 0,& \text{otherwise,}\end{cases}$$

Solving for marginal PDF:

$$f_X(x) = \int\limits_{-\infty}^\infty f_{X,Y}(x,y)\,dy = \int\limits_y f_{X,Y}(x,y)\,dy = \int\limits_0^x 2\,dy = 2\cdot y\Big |_0^x = 2x$$

Thus,

$$f_{X}(x) = 2x\textrm{ for } x\in[0,1]$$

• I think you're right, I deleted my post. – Erik Cristian Seulean Mar 24 at 15:02
• Hey, you have been great help. Thanks! – muxo Mar 25 at 6:58