Definition:Sub-Gaussian Distribution

Definition
The distribution of a random variable $X$ with mean $\mu=\mathbb E\left[X\right]$ is called sub-Gaussian if there exists a $\sigma>0$ such that

\begin{equation}\mathbb{E}\left[e^{\lambda(X-\mu)}\right] \leq e^{\sigma^{2} \lambda^{2} / 2}\end{equation}

for all $\lambda \in \mathbb{R}$.