exponential distribution question

Let $\displaystyle Y$ be an exponentially distributed random variable with mean $\displaystyle \beta$. Define a random variable $\displaystyle X$ in the following way: $\displaystyle X=k$ if $\displaystyle k-1 \leq Y \leq k \ \ \mbox{for} \ \ k =1,2,...,n$

**a)** Find $\displaystyle P(X=k) \ \ \forall \ \ k=1,2,...,n$

**b)** Show that your answer to part **a)** can be written as:

$\displaystyle P(X=k) =(e^{-\frac{1}{\beta}})^{k-1}(1-e^{-\frac{1}{\beta}}) \ \ \forall \ \ k =1,2,...,n$

I would think for **a)** it would be $\displaystyle \int_{k-1}^{k} \frac{1}{\beta} e^{-\frac{x}{\beta}} dy = -e^{-\frac{x}{\beta}} \bigg{|}^{k}_{k-1} = -e^{-\frac{k}{\beta}}+e^{-\frac{k-1}{\beta}}$, but this doesn't seem right since k is not continuous.

and for **b)** I would think that you have to manipulate the function you got in **a)** to get what is shown, but am clueless on how to do so. Any help would be greatly appreciated.