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:

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


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

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  • 1
    $\begingroup$ I think you're right, I deleted my post. $\endgroup$ – Erik Cristian Seulean Mar 24 at 15:02
  • 1
    $\begingroup$ Hey, you have been great help. Thanks! $\endgroup$ – muxo Mar 25 at 6:58

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