<s>[INST] Conditional Joint Distribution Concern Random variables $X$ and $Y$ are jointly distributed with density 
$$f(x,y)=
\begin{cases}
3x & 0 <y<2x<2 \\
0 & \text{otherwise}
\end{cases} 
$$
(i) Find $P\left(X<\frac{1}{2} \Big| Y < \frac{1}{2}\right)$
$$
P\left(X<\frac{1}{2} \Big| Y < \frac{1}{2}\right) = \frac{P\left(X < \frac{1}{2} \cap Y < \frac{1}{2} \right)}{P\left( Y < \frac{1}{2}\right)} \\ = \frac{ \int_0^{\frac{1}{2}} \int_{\frac{y}{2}}^{\frac{1}{2}} 3x \, dx\,dy}{\int_0^{\frac{1}{2}} \int_{\frac{y}{2}}^{2} 3x \, dx\,dy}$$
Would someone help me out in verifying my work and if it has anything wrong to direct me to the right way. 
 [/INST] In the denominator you should have
$\int_{0}^{\frac{1}{2}} \int_{\frac{y}{2}}^{1} 3x dx dy$
and in the numerator you should have
$\int_{0}^{\frac{1}{2}} \int_{\frac{y}{2}}^{\frac{1}{2}} 3x dy dx$
In other words, you should have $$\dfrac{\int_{0}^{\frac{1}{2}} \int_{\frac{y}{2}}^{\frac{1}{2}}3x dx dy}{\int_{0}^{\frac{1}{2}} \int_{\frac{y}{2}}^{1} 3x dx dy}$$
</s>