Independent solution

How to solve this Exponential Distribution question

Setup

Setup

A capped reimbursement leaves part of a loss unpaid exactly when the loss exceeds the cap.

pA=Pr⁡(X>m),pB=Pr⁡(X>m+2)p_A=\Pr(X>m),\qquad p_B=\Pr(X>m+2)

Model

Model

Apply the survival function for an exponential variable with mean six.

pA=e−m/6,pB=e−(m+2)/6p_A=e^{-m/6},\qquad p_B=e^{-(m+2)/6}

Compute

Compute

Form the requested ratio and cancel the common cap term.

pBpA=e−(m+2)/6e−m/6\frac{p_B}{p_A}=\frac{e^{-(m+2)/6}}{e^{-m/6}}
pBpA=e−2/6=e−1/3\frac{p_B}{p_A}=e^{-2/6}=e^{-1/3}

Answer

Answer

The ratio is exp(-1/3), approximately 0.7165.

e−1/3(B)\boxed{e^{-1/3}\quad\text{(B)}}