Independent solution

How to solve this Beta Distribution question

Setup

Setup

With the stated beta parameters, the density is proportional to r¹²(1-r). Normalize it on the unit interval.

1=k∫01(r12−r13) dr=k(113−114)1=k\int_0^1(r^{12}-r^{13})\,dr=k\left(\frac1{13}-\frac1{14}\right)
k=182k=182

Model

Model

The requested upper-tail probability is the integral of the normalized density above the threshold.

Pr⁡(R>0.9)=182∫0.91(r12−r13) dr\Pr(R>0.9)=182\int_{0.9}^{1}(r^{12}-r^{13})\,dr

Compute

Compute

Use the polynomial antiderivative and evaluate it at both endpoints.

Pr⁡(R>0.9)=[14r13−13r14]0.91\Pr(R>0.9)=\left[14r^{13}-13r^{14}\right]_{0.9}^{1}
Pr⁡(R>0.9)=1−14(0.9)13+13(0.9)14\Pr(R>0.9)=1-14(0.9)^{13}+13(0.9)^{14}
Pr⁡(R>0.9)=0.4153708595\Pr(R>0.9)=0.4153708595

Answer

Answer

The upper-tail probability rounds to 0.415.

0.415(C)\boxed{0.415\quad\text{(C)}}