I am currently dealing with the joint density function of continuous random variables.

Here's the following piece-wise function I'm working on:

$$f(x,y)= \begin{cases} 24x & \text{if } 0 \le y \le 1-2x \text{ and } 0 \le x \le .5 \\ 0 & \text{otherwise} \end{cases}$$

Support of this pdf (not to scale): support using the bounds

How could I rearrange the bounds so that I could satisfy all of the y values in terms of x for the marginal PDF of random variable X?


1 Answer 1


The domain of interest is a triangle bounded by the $x$-axis, $y$-axis and the line $y=1-2x$.

Look at the what values does $y$ value take over the region. I will leave it as an exercise to figure out the two numbers, $c_1$ and $c_2$.

Once we fix the value of $y$, notice that $x$ can take any value between $0$ and $x=\frac{1-y}2$.

Hence $0 \le x \le \frac{1-y}2$ and $c_1 \le y \le c_2$,


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