这是一份warwick华威大学ST117-15的成功案例

If a quantile function $S(p)$, has a standard distribution in the sense that some measure of position, e.g., the median, is zero and some linear measure of variability, scale, e.g., the $I Q R$, is one, then the quantile function
$$
Q(p)=\lambda+\eta S(p)
$$
has a corresponding position parameter of $\lambda$ and a scale parameter of $\eta$.
Thus the two parameters $\lambda$ and $\eta$ control the position and spread of the quantile function. Obviously if we know the parameters of a quantile function, we can use this result in reverse to create a standard distribution:
$$
S(p)=[Q(p)-\lambda] / \eta
$$

ST117-15 COURSE NOTES ：

$$
E\left(x^{2}\right)=\int_{0}^{1}\left[1 /(1-p)^{2 \beta}\right] d p
$$
By analogy with the previous calculation, this is seen to be
$$
\mu_{2}^{\prime}=1 /(1-2 \beta), \text { for } 0<\beta<0.5 \text {. }
$$
Notice that we now have an even tighter constraint to give a finite variance. On substituting in the expression for $\mu_{2}$ and simplifying, we finally obtain
$$
\mu_{2}=\beta^{2} /\left[(1-2 \beta)(1-\beta)^{2}\right] \text {, for } 0<\beta<0.5
$$