Independent solution

How to solve this Continuous Random Variables question

Setup

Setup

Write the density on the left half as c divided by t+1. Symmetry makes its total probability one half.

f(t)=ct+1,0≤t≤6.5f(t)=\frac{c}{t+1},\qquad 0\le t\le6.5
∫06.5ct+1 dt=cln⁡(7.5)=0.5\int_0^{6.5}\frac{c}{t+1}\,dt=c\ln(7.5)=0.5
c=0.5ln⁡(7.5)c=\frac{0.5}{\ln(7.5)}

Model

Model

Let k be the 60th percentile. Its reflection 13-k is the 40th percentile and lies on the known left half.

FT(13−k)=1−FT(k)=0.40F_T(13-k)=1-F_T(k)=0.40

Compute

Compute

Integrate the left density to the reflected point, then solve for k.

0.40=0.5ln⁡(7.5)ln⁡(14−k)0.40=\frac{0.5}{\ln(7.5)}\ln(14-k)
ln⁡(14−k)=0.8ln⁡(7.5)\ln(14-k)=0.8\ln(7.5)
k=14−7.50.8=8.9875620k=14-7.5^{0.8}=8.9875620

Answer

Answer

The student registers after approximately 8.99 elapsed days.

8.99days (C)\boxed{8.99\quad\text{days (C)}}