We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ are known in advance.

Define $S_n=\sum_{i=1}^nX_i$.

As we can see $\boldsymbol X$ can be viewed as inter-arrival times for a renewal process, and $S_n$ denotes each arrival epoch.

Next we define a variable $K$: $K=\inf\, \{n\mid S_n>T\}$ (or $K=\min\, \{n\mid S_n > T\}$), where $T$ is a constant.

Then

- what is the distribution of $K$?
- what is the distribution of $S_K$?

I already know that the PDF for $S_n$, denoted by $f_n$, can be computed by $f_n=f^{*n}=f*f*\cdots *f$, the $n$-fold convolution power of $f(x)$. By Laplace Transform, we can convert the convolution to multiplication.