Since the cdf $F$ is a monotonically increasing function, it has an inverse; let us denote this by $F^{−1}$. If $F$ is the cdf of $X$, then $F^{−1}(\alpha)$ is the value of $x_\alpha$ such that $P(X \le x_\alpha) = \alpha$; this is called the $\alpha$ quantile of $F$. The value $F^{−1}(0.5)$ is the median of the distribution, with half of the probability mass on the left, and half on the right. The values $F^{−1}(0.25)$ and $F^{−1}(0.75)$ are the lower and upper quartiles. We can also use the inverse cdf to compute tail area probabilities. For example, if $\Phi$ is the cdf of the Gaussian distribution $N (0, 1)$, then points to the left of $\Phi^{−1}(\alpha)/2)$ contain $\alpha/2$ probability mass, as illustrated in Figure 2.3(b). By symmetry, points to the right of $\Phi^{−1}(1−\alpha/2)$ also contain α/2 of the mass. Hence the central interval $(\Phi^{−1}(\alpha/2), \Phi^{−1}(1 − \alpha/2))$ contains $1 − \alpha$ of the mass. If we set $\alpha = 0.05$, the central $95\%$ interval is covered by the range $(\Phi−1(0.025), \Phi−1(0.975)) = (−1.96, 1.96)$ (2.23) If the distribution is $N (\mu, \sigma^2)$, then the $95\%$ interval becomes $(\mu − 1.96\sigma, \mu + 1.96\sigma)$. This is sometimes approximated by writing $\mu \pm 2\sigma$.

Could you explain all this with an example?