Help me understand the following equality $\mathbb{E} [e^{\lambda a_i g_i}] = e^{\lambda^2 a_i^2/2}$

Question

we are given a standard Gaussian random variable $g_i$ and a real value $0<=a_i<=1$ and $\lambda \in \mathbb{R}$. Why is $\mathbb{E} [e^{\lambda a_i g_i}] = e^{\lambda^2 a_i^2/2}$?

Thank you very much in advance!

Answer 1

Suppose that $X \sim \mathcal{N}(0,1)$. Then, for all $t\in\mathbb{R}$, $$\mathbb{E}[e^{t X}]=\int_{-\infty}^\infty e^{t x}f_X(x)\:dx=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{t x}e^{-x^2/2}\:dx=\frac{e^{t^2/2}}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-{(x-t)^2/2}}\:dx=e^{t^2/2}.$$ Therefore, for all $\lambda, a \in \mathbb{R}$, $$\mathbb{E}[e^{\lambda aX}]=e^{\lambda^2a^2/2}.$$