Expected Value of Exponential

Question

I am attempting to calculate this expected value and am having some serious trouble:

$$ \mathbb{E}\left[\,\exp\left(\vphantom{\LARGE A}\,\beta_{1}\left(\vphantom{\Large A}\,\left\vert\, z - \lambda\,\right\vert - \mathbb{E}\left[\vphantom{\large A}\,\left\vert\, z - \lambda\,\right\vert\,\right]\,\right)\,\right)\,\right] $$ where $z \sim \mathcal{N}(0,1)$.

Normally I would be alright with these sort of calculations but the nesting of an expected value in an exponential with an expected value has thrown me off. Any help or guidance would be greatly appreciated.

Answer 1

To deal with the formula in your own comment. The PDF of $z$ is $f(x)=\frac{1}{\sqrt{2*\pi}}*e^{-1/2*x^2}$ and $E(x)=\int xf(x)dx$. So $E(e^{\lvert{x}\rvert})=\int e^{\lvert{x}\rvert} f(x) dx$=$e^{1/2}$. I may have made a mistake in the calculation, but this gives you an idea of how to proceed.

Answer 2

Hint:

Use Taylor series for $e^{\beta_1 |z-\lambda|}$

For $e^x$, the taylor series is $= 1+x+\frac{x^2}{2!}+...$

Replace x with $\beta_1 |z-\lambda|$

Then $\mathbb{E}\left[{e^{\beta_1 |z-\lambda|}}\right] = 1+ \mathbb{E}\left[{\beta_1 |z-\lambda|}\right]+ \mathbb{E}\left[{\frac{(\beta_1 |z-\lambda|)^2}{2}}\right] + O(\frac{1}{n^2})$

Which is easier to deal with.