Let X be an exponential(1) random variable, and define Y to be the integer part of $X+1$, that is $Y=i+1$ if and only if $i \leq X < i+1, i=0,1,2,...$
Find the conditional distribution of $X-4$ given $Y \geq 5$.
My attempt: $Y \geq 5$ is equivalent $X \geq 4$. Thus $P(X-4 \leq x | Y \geq 5)=P(X-4 \leq x | X \geq 4) = \frac{P(4 \leq X \leq x+4)}{P(X \geq 4)} = 1-e^{-x}$
I don't know whether my answer is correct.
Your answer is correct (although I assume you mean $\mathbb{P}(4\le\mathsf{X}\le x+4)$ in the numerator in your third expression) and it neatly shows a defining feature of the exponential distribution, namely that it is memoryless, i.e. $\mathbb{P}(\mathsf{X}\ge x+k\mid\mathsf{X}\ge k)=\mathbb{P}(\mathsf{X}\ge x)$, meaning that the conditional distribution of $\mathsf{X}-x$ given $\mathsf{X}\ge x$ is just the distribution of $\mathsf{X}$.