Find the characterisitc function for a standardised binomial random variable

Question

Find the characteristic function of $Y$.

I know the formula for the characteristic function. This is a question from a past exam that doesn't provide answers and I'm unsure where to start. Any help would be greatly appreciated. Sorry if formatting isn't correct!

Answer 1

We assume you know $\phi_X(t)=E(e^{itX})=\sum_1^n \binom{n}{k} e^{itk} p^k(q)^{n-k}=\sum \binom{n}{k} (e^{it}p)^k (q)^{n-k}=(pe^{it}+(q))^n$.

Now can you work out a way to get $E(e^{itX})$ from $E(e^{itY})$?

More so can you factor: $$E(e^{itY})$$ After that we are done.

Well we know $$E(e^{itY})=E(e^{it\frac{X-np}{\sqrt{npq}}})=E(e^{it\frac{X}{\sqrt{npq}}}e^{-it\frac{np}{\sqrt{npq}}})=\phi_X \left(\frac t{\sqrt{npq}}\right)e^{-it\frac{np}{\sqrt{npq}}}$$

And ${\sqrt{npq}}$ is a constant.

How about now?

Answer 2

In general $$\phi_{aX+b}(t)=\mathbb Ee^{it(aX+b)}=\mathbb Ee^{itb}e^{iatX}=e^{itb}\mathbb Ee^{iatX}=e^{itb}\phi_X(at)$$