I a trying to understand the following result from this lecture notes. Defining the value at risk as a function of the random variable $L$ and the parameter $\alpha \in (0,1)$:
$$ \operatorname{VaR}_{\alpha}(L):=q_{\alpha}(L)=\inf \left\{x \in \mathbb{R}: F_{L}(x) \geq \alpha\right\} $$
It is stated that:
Consider two IID assets, $X$ and $Y$ where $$ \begin{aligned} X=\epsilon+\eta \quad \text { where } \quad \epsilon & \sim \mathrm{N}(0,1) \\ & \text { and } \quad \eta=\left\{\begin{array}{ll} 0, & \text { with prob } .991 \\ -10, & \text { with prob } .009 \end{array}\right. \end{aligned} $$ Consider a portfolio consisting of $X$ and $Y$. Then $$ \begin{aligned} \operatorname{VaR}_{.99}(X+Y) &=9.8 \\ &>\operatorname{VaR}_{.99}(X)+\operatorname{VaR}_{.99}(Y) \\ &=3.1+3.1 \\ &=6.2 \end{aligned} $$
I know how to compute $VaR_\alpha(L)$ for a normally distributed rv. In this case, how is $VaR_{0.99}(X)$ computed?, can we obtain the CDF of $X$?
The cumulative distribution function of $X$ is $$P(X \le x)=P(\eta = 0)P(\epsilon \le x) + P(\eta = -10)P(\epsilon \le 10+x) \\ = 0.991 \Phi(x) +0.001 \Phi(10+x)$$
so the $\mathrm{VaR}_{0.99}(X) \approx 3.1$ comes from $$P(X \le -3.1) \approx 0.991\times 0.001 + 0.009 \times 1 \\\approx 0.01 \\=1-0.99$$