Mixture of random variables

Question

I'm trying to find the distribution of a mixtures of random variables.

Given $ X\thicksim \varepsilon(\lambda)$ (an exponential distribution) and $Y=g(X)+1$ , where g(X) means the integer part of X. What's the distribution of Y? The solution is: "Y has a geometric distribution with $1-e^{-\lambda}$ as parameter.

Answer 1

I don't really know what you mean by mixture. The +1 is harmless, it just shifts the support to $\{ 1,2,\dots, \}$ instead of $\{ 0,1,\dots \}$. With that in mind, you just need to compute $P(g(X)=n)=P(X \in [n,n+1))=\int_n^{n+1} \lambda e^{-\lambda x} dx$ for $n=0,1,\dots$.