Expected value of a r.v. with respect to its mode

Question

Let $X$ be a random variable having an exponential distribution such that

$E[X]=\frac {1}{10}$

Which is the probability that $X$ is smaller than its mode $\nu$?

My intuition is that this probability is equal to $0$ since $\frac {1}{10}$ is in the first quadrant and the function goes to $+\infty$ but I am not sure if I have interpreted the question well.

Answer 1

You are right. The mode is zero because the value of the pmf is maximum there. The requested probability then, is 0 since the r.v. only takes over non negative real numbers