$X$ is a random variable. If I am given the $P(X<1)= 0$ and the $P(X>e) = 0$ and in the range $y \in [1,e]$ the $P(X<y)= \int_1^y \frac{1}{x}\; dx$. If $\mu$ is the expected value of $X$, what is the closest integer to $10 k$, where $k$ is given by $P(X- \mu>k)=0.05$.

My approach was to calculate the $\mu$ of $X$, which I think is the integral where $y = e$. What I get is $1/(e-1)$, which is $0.58$.

Does anyone know where to go from here? I thought to manipulate $P(X- \mu>k)=0.05$ to make it $1-P(X- \mu<k)=0.05$, because I have a given expression from here.