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I have been workin on some problems involving conditional probability and continuous random variables and the thing is i dont know if i get the limits correct, anyway here is the problem, check it out, any suggestions would be helpful

[tex]f(y_1,y_2) =\left\{\begin{array}{cc}2,&\mbox{ if }

0\le y_1\le 1, 0\le y_2\le 1, y_1+y_2\le 1\\0, & \mbox{elsewhere}\end{array}\right[/tex]

what i want to find was

[tex]P(Y_1\ge \frac{1}{2}|Y_2\le \frac{1}{4})[/tex]

[tex]=\frac{\int_{0}^{\frac{1}{4}} \int_{\frac{1}{2}}^{1-y_2} 2 dy_1 dy_2}{\int_{0}^{\frac{1}{4}} \int_{0}^{1-y_2} 2 dy_1 dy_2}=\frac{3}{7}[/tex]

also I wanted to find

[tex]P(Y_1\ge \frac{1}{2}|Y_2=\frac{1}{4})[/tex]

[tex]=\frac{\int_{\frac{1}{2}}^{\frac{3}{4}} 2 dy_1}{\int_{0}^{1} 2 dy_1}=\frac{1}{4}[/tex]

about the 3/4 that is where the intersection occurs

now have i got my limits correct? how do i know if i have the limits correct? are my answers corrrect?

steven