Figure 8.

(Left) A time series illustration of how a higher variance (hence scale), given the same mean, allow more spikes, hence an antifragile effect. We have random paths of two gamma distributions of the same mean, different variances, $X_1\sim G(1,1)$ and $X_2\sim G(\frac{1}{10},10)$, showing higher spikes and maxima for $X_2$. The effect depends on the norm ${\left|\right|.\left|\right|}_{\infty }$, more sensitive to tail events, even more than just the scale which is related to the norm ${\left|\right|.\left|\right|}_2$. (Right) Representation of antifragility of (Left) in distribution space: we show the probability of exceeding a certain threshold for a variable, as a function of $\sigma$, the scale of the distribution, while keeping the mean constant.