Prove this property of characteristic function

Question

I have been trying to prove this for some time now but I have reached nowhere.

$\phi(X)$ is the characteristic function of rv X. Prove: $\phi(X+Y) = \phi(X)\phi(Y)$

if X and Y are independent rv. I tried expanding them as E[$e^{itX}$] but that doesn't lead me anywhere. Please help.

It would help to know what you're allowing yourself to start from, as this can either be very easy or quite a mess depending on what theorems we may invoke as justification. Note that the question can easily be reduced to "show that the Fourier transform turns convolution into multiplication." — user3716267, Sep 08 '21 at 18:40
Basic definition has been taught to me yet. So i guess its the very easy part maybe — Alex, Sep 08 '21 at 18:42

score 1 · Accepted Answer · answered Sep 08 '21 at 18:37

1

$\phi(X+Y) = E[e^{it(X+Y}] = E[e^{itX}e^{itY}] = E[e^{itX}]E[e^{itY}] = \phi(X) \phi(Y)$.

answered Sep 08 '21 at 18:37

Alvin Jin

1 Answers1