Show that $\lambda_1 S_1$ is exponential of parameter 1.

Question

In Norris his book about Markov Chains the following question pops up:

Let $S_1, S_2\dots$ be independent exponential random variables with parameters $\lambda_1, \lambda_2 \dots$ respectively. Show that $\lambda_1 S_1$ is exponential of parameter $1$.

I do not understand this question what is meant by "exponential of parameter $1$"? What do I have to show?

Answer 1

An exponential variable S, by definition, has a density of the form

\lambda \exp(-\lambda x)

where \lambda is an arbitrary positive number. This number is so characteristic of the variable that it is called the "parameter".

The assignment invites you to demonstrate that \lambda S is another exponential variable, presumably by calculating its density function.