Problem: The lifetimes of elements of a certain type are independent and exponentially distributed with parameter $\lambda > 0$. At time $t= 0$ there are $X_0=n$ living elements present. Let $X_t$ denote the number alive at time $t$. Show that $\{X_t\}$ is a Markov process and calculate its transition probabilities.
Let $X_t = n-Y_t$, $Y_t$ be the number of elements that died, I think $Y_t$ is a Poisson process, but I'm not sure. If it is, the rate must then depend on the state. For example if there are $n$ elements alive, the rate is $n \lambda$, but then I'm stuck, how do I find the transition probabilities?