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(α) is the value of xα such that P(X ≤ xα) = α; this is called the α 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 Φ is the cdf of the Gaussian distribution N (0, 1), then points to the left of Φ−1(α)/2) contain α/2 probability mass, as illustrated in Figure 2.3(b). By symmetry, points to the right of Φ−1(1−α/2) also contain α/2 of the mass. Hence the central interval (Φ−1(α/2), Φ−1(1 − α/2)) contains 1 − α of the mass. If we set α = 0.05, the central 95% interval is covered by the range (Φ−1(0.025), Φ−1(0.975)) = (−1.96, 1.96) (2.23) If the distribution is N (μ, σ2), then the 95% interval becomes (μ − 1.96σ, μ + 1.96σ). This is sometimes approximated by writing μ ± 2σ.

Could you explain all this with an example?