Condition Distribution of $~X~$ given $~X<2Y~$

Question

Suppose that $~X,~ Y~$ are independent exponential random variables with parameter $\lambda$.

For $a > 0$, find $P (X > a|X < 2Y )$. What is the conditional distribution (give the name and any parameter(s)) of $X$ given that $X < 2Y$ ?

Attempt

I have calculated that $P (X > a|X < 2Y ) = \frac{3 e^{-\lambda a}-1}{2}$.

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Question

What distribution does $P (X > a) = \frac{3 e^{-\lambda a}-1}{2}$ represent?