Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Let beta denote the exponential mean and Y the loss limited at the reimbursement maximum.

β=400ln⁡2,Y=min⁡(X,1000)\beta=\frac{400}{\ln 2},\qquad Y=\min(X,1000)

Model

Model

Use the survival-integral identity for a nonnegative variable capped at a fixed amount.

E[Y]=∫01000Pr⁡(X>x) dxE[Y]=\int_0^{1000}\Pr(X>x)\,dx
Pr⁡(X>x)=e−x/β\Pr(X>x)=e^{-x/\beta}

Compute

Compute

Integrate the exponential survival and simplify the exponent.

E[Y]=β(1−e−1000/β)E[Y]=\beta\left(1-e^{-1000/\beta}\right)
1000β=1000ln⁡2400=2.5ln⁡2\frac{1000}{\beta}=\frac{1000\ln 2}{400}=2.5\ln 2
E[Y]=400ln⁡2(1−2−2.5)=475.0640717E[Y]=\frac{400}{\ln 2}\left(1-2^{-2.5}\right)=475.0640717

Answer

Answer

The expected limited payment rounds to 475.

475(D)\boxed{475\quad\text{(D)}}