Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Let T have exponential rate λ, let m be its median, and let mu be its mean.

S(t)=e−λtS(t)=e^{-\lambda t}
S(m)=12,μ=1λS(m)=\frac{1}{2},\qquad \mu=\frac{1}{\lambda}

Model

Model

The probability of lying between two endpoints is the difference of their survival probabilities.

Pr⁡(m<T<μ)=S(m)−S(μ)\Pr(m<T<\mu)=S(m)-S(\mu)

Compute

Compute

Use the defining survival probability of the median and substitute the exponential mean into the other endpoint.

S(μ)=e−λ(1/λ)=e−1S(\mu)=e^{-\lambda(1/\lambda)}=e^{-1}
Pr⁡(m<T<μ)=12−e−1=0.1321205588\Pr(m<T<\mu)=\frac{1}{2}-e^{-1}=0.1321205588

Answer

Answer

The interval probability is one half minus e to the negative one.

12−e−1(A)\boxed{\frac{1}{2}-e^{-1}\quad\text{(A)}}