# What happen if a probability function does not converge?

Find the marginal distribution for $$y_2$$ given the following PDF

$$f(y_1,y_2)= \begin{cases} 3y_1, & \text{if } 0\leq y_2\leq y_1 \\ 0, & \text{elsewhere} \end{cases}$$

But when I try to find it, I end up with the following integral:

$$f_{Y_2}(y_2)=\int_{y_2}^\infty 3y_1 \, dy_1$$

and that integral does not converge. Is there any trick to solve this? or is this situation an special case?

Thanks

• You should have $f_{Y_2}(y_2)$ where you have $f_{y_2}(y_2). \qquad$ Commented Aug 26, 2020 at 23:17
• The function that you called the PDF is not in fact a probability density function, for just the reason you stated. Commented Aug 26, 2020 at 23:18
• This is not a PDF as it stands. I suspect you mean $0 \le y_2 \le y_1 \le 1$. Commented Aug 26, 2020 at 23:19

Assuming you actually meant $$0, first check that you have pdf by integrating: $$\int_0^1 \int_0^{y_1} 3y_1 \, dy_2 \, dy_1$$ Once you've done that, marginalize out $$Y_2$$: $$f_{Y_2}(y_2) = \int_{y_2}^1 3y_1 \, dy_1$$ Again, check this by integrating in the $$[0,1]$$ interval to get $$1.$$
• No, in my homework is $0\leq y_2\leq y_1$. I think the way my teacher write the homework is wrong Commented Aug 26, 2020 at 23:49
• Try integrating the joint pdf, and see if you can get $1$. If not, there's an error somewhere