Independent solution

How to solve this Normal Distribution question

Setup

Setup

Let a be the loss CDF evaluated at the deductible and use the supplied CDF value at the higher threshold.

a=FX(10000),FX(k)=0.9582a=F_X(10000),\qquad F_X(k)=0.9582

Model

Model

Write the conditional probability as the mass between the deductible and the higher threshold divided by the mass above the deductible.

0.9582−a1−a=0.9500\frac{0.9582-a}{1-a}=0.9500

Compute

Compute

Solve for the deductible percentile, standardize it, and recover the unknown standard deviation.

0.9582−a=0.95(1−a),a=0.1640.9582-a=0.95(1-a),\qquad a=0.164
Φ−1(0.164)=−0.9781503\Phi^{-1}(0.164)=-0.9781503
10000−12000c=−0.9781503\frac{10000-12000}{c}=-0.9781503
c=2044.6756c=2044.6756

Answer

Answer

The standard deviation rounds to 2045.

2045(A)\boxed{2045\quad\text{(A)}}