Independent solution

How to solve this Continuous Distributions question

Setup

Setup

Write the density kernel using a fractional exponent and determine its normalizing constant.

f(x)=cx1/n,0<x<1f(x)=c x^{1/n},\qquad 0<x<1
1=c∫01x1/n dx=cnn+11=c\int_0^1x^{1/n}\,dx=c\frac{n}{n+1}
c=n+1nc=\frac{n+1}{n}

Model

Model

Integrate the normalized density and invert its power CDF at a general percentile level.

F(x)=x(n+1)/nF(x)=x^{(n+1)/n}
qp=pn/(n+1)q_p=p^{n/(n+1)}

Compute

Compute

Take the ratio of the two quantiles and combine their percentile levels.

q0.30q0.20=(0.300.20)n/(n+1)\frac{q_{0.30}}{q_{0.20}}=\left(\frac{0.30}{0.20}\right)^{n/(n+1)}
q0.30q0.20=(1.5)n/(n+1)\frac{q_{0.30}}{q_{0.20}}=(1.5)^{n/(n+1)}

Answer

Answer

The ratio is the positive (n+1)st root of 1.5 to the nth power.

(1.5)nn+1(D)\boxed{\sqrt[n+1]{(1.5)^n}\quad\text{(D)}}