Conditional distribution of $X$ given $Y$ I'm practicing on conditional distribution. This is my exercise

Suppose that the lifespans of two light bulbs are independent and identically distributed exponential RVs. Let $X$ and $Y$ be the times at which the first and second bulbs burn out, respectively ($X \leq Y$). What is the conditional distribution of $X$ given $Y$.

Can I do something like this: Since $X,Y$ are independent, then $f_{X,Y}(x,y) = f_X(x).f_Y(y)$. Then $f_{X}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)} = f_X(x)$. I think that I must be so wrong about this. Can somebody please help me out.
 A: Let the lifespans of the bulbs be $A$ and $B$. These are iid exponential RVs. Then $X = \min\{A, B\}$ and $Y = \max\{A, B\}$. When I put it like that, maybe it's easier to see why $X$ and $Y$ are not independent, which is where your attempted solution breaks.
To actually solve the problem, we can try to understand the joint distribution of $X$ and $Y$. Suppose $A, B \sim Exp(\lambda)$. For $s \le t$ the joint CDF will be
$$\begin{align}
F_{X,Y}(s,t) &= P[X \le s, Y \le t] \\
&= P[A \le t \text{ and } B \le t] - P[s < A \le t \text{ and } s < B \le t] \\
&= F_A(t) F_B(t) - [F_A(t)-F_A(s)][F_B(t)-F_B(s)] \\
&= \left( 1 - e^{-\lambda t}\right)^2 - \left( e^{-\lambda s} - e^{-\lambda t}\right)^2
\end{align}$$
Then for the joint pdf we get
$$\begin{align}
f_{X,Y}(s,t) &= \frac{d^2}{dsdt} F_{X,Y}(s,t) \\
&= 2 \lambda^2 e^{-(s+t)\lambda}
\end{align}$$
Also, the CDF for $Y$ is $F_Y(t) = F_{X,Y}(t, t)$, so the pdf is
$$\begin{align}
f_{Y}(t) &= \frac{d}{dt} \left( 1 - 2 e^{-\lambda t} + e^{-2 \lambda t} \right) \\
&= 2 \lambda \left( e^{-\lambda t} - e^{-2 \lambda t} \right)
\end{align}$$
Now we're finally ready to plug into the formula for conditional pdf:
$$\begin{align}
f_{X|Y}(x|y) &= \frac{f_{X,Y}(x,y)}{f_{Y}(y)} \\
&= \frac{2 \lambda^2 e^{-(x+y)\lambda}}
{2 \lambda \left( e^{-\lambda y} - e^{-2 \lambda y} \right)} \\
&= \boxed{\frac{\lambda e^{\lambda(t-s)}}{e^{\lambda t} - 1}}
\end{align}$$
Note that the final formula, like all the other cdfs and pdfs computed in this solution, is only valid on the region $0 \le s \le t$. Outside of that region the pdf is 0.
