Quantiles of monotone functions of rvs. Here, QX (α) and QY (α) will denote the α-th quantiles of the distributions of X and Y , respectively.
1. Let Y be a Gamma(n, β) rv, where n is a positive integer. Let QX (α) be the quantile function of the χ2 with 2n degrees of freedom distribution. Express QY (α) in terms of QX (α).
2. Let Y be a N ormal(µ, σ2) rv. If QX (α) = Φ^-1(α) is the quantile function of a Normal(0, 1) distribution with the cdf Φ, express the quantile function of the distribution of Y in terms of QX (α).
3. Let h be a strictly decreasing continuous function and X be a continuous rv with the quantile function QX . Let Y = h(X). Express the quantile function QY in terms of the quantile function QX . As a sanity check, you can choose any tranctable distribution that you like and h(x) = a + bx where b < 0 in order to see if your general answer makes sense.