What are the broad steps required to solve a question like this?

Let $Y=aX+b$, where $X\sim\text{Exp}(\lambda)\,,\:\lambda>0$ and find the characteristic function of $Y$.

  • $\begingroup$ please add the self-study tag and read its tag wiki $\endgroup$ – Glen_b Oct 11 '16 at 11:28

There's a couple of ways to go.

Note that the characteristic function (c.f.) is an expectation, from its definition. Start by writing it in that form.

One approach would be to use properties of expectation to pull out a multiplicative term in $b$ and have a c.f. of $aX$; then if $X$ is exponential, you can write $aX$ as another exponential variable, and if you know the c.f. of an exponential you can immediately write down the cf of $aX$ - and hence (via the multiplicative constant that was taken out of the expectation before), the c.f. of $aX+b$.

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  • $\begingroup$ i'm unfamiliar with what an a.c.f. actually is. What's the other way of doing this question as I might be more prepared to use that method. $\endgroup$ – gorge Oct 11 '16 at 12:17
  • $\begingroup$ @gorge To my knowledge I never mentioned "a.c.f." in my answer (if I did it's a typo)... I wrote "c.f." -- the usual abbreviation for "characteristic function". The initial letters c.f. are commonly used to refer to the characteristic function (just as m.g.f. refers to the moment generating function). In any case, I will edit to make the abbreviation explicit after the first mention of "characteristic function" (which I spelled out in full on the first use) $\endgroup$ – Glen_b Oct 11 '16 at 23:13
  • $\begingroup$ +1 I've been tempted to unfold the entire answer, but I'm sure you looked at the OP as an exercise, and provided enough guidance. Ty $\endgroup$ – Antoni Parellada Oct 12 '16 at 17:53
  • $\begingroup$ @Antoni In this case OP explicitly asked for "broad steps" (in contrast to a request for an explicit solution) which is what I attempted to outline, so it wasn't necessary to try to decide whether it was appropriate to say more. However, I have decided to add just a teensy bit more detail in that. $\endgroup$ – Glen_b Oct 12 '16 at 21:23

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