Problem: The life expectancy $X$ of a lightbulb can be modeled as an exponential distribution with $\lambda > 0$. Every started hour shall be counted fully. Therefore consider $K:=\lceil X \rceil $ and distribution of $K$.
I'm not quite sure I understand the question correctly, so I wanted to ask for some advice concerning the topic. I kind of think of it as a discrete modeled distribution since we are always looking at $\lceil X \rceil $. So here it would be: $$P(K = t)=P(\lceil X \rceil = t)=P(t-1<X\le t)$$ Is this a correct way of approaching the problem? If yes where do I go from here? Thanks in advance.
I guess you mean $K$ not $T$ in the beginning. Yes, indeed, $$ \mathbb{P}[K=t] = \mathbb{P}[t-1 < X \le t] = F_X(t) - F_X(t-1). $$ Can you find the cdf of the exponential distribution?