Independent solution

How to solve this Exponential Distribution question

Setup

Setup

A policy makes a positive payment precisely when the loss exceeds its deductible, so the supplied probabilities are survival values.

S(x)=e−λxS(x)=e^{-\lambda x}
S(1.44)=0.640,S(d)=0.512S(1.44)=0.640,\qquad S(d)=0.512

Model

Model

Taking logarithms removes the unknown rate and relates the two thresholds directly.

d1.44=log⁡(0.512)log⁡(0.640)\frac{d}{1.44}=\frac{\log(0.512)}{\log(0.640)}

Compute

Compute

The two survival values are consecutive powers of 0.8, which makes the ratio especially transparent.

0.640=0.82,0.512=0.830.640=0.8^2,\qquad 0.512=0.8^3
d=1.44(32)=2.160d=1.44\left(\frac32\right)=2.160

Answer

Answer

The deductible associated with the smaller exceedance probability is 2.160.

2.160(E)\boxed{2.160\quad\text{(E)}}