If $\lambda$ defines how long it takes until we see an event occurring in a one-time unit- does the following hold true:
Let's say $\lambda=2$ for "$1$ bus passing by every $2$ seconds" so $1/\lambda = 1/2$ means it is expected for us to wait for $1/2$ of a second in order to see one bus passing by?
So $E[X] = 1/\lambda$ expresses the amount of time we have to wait on avg. to see an event occurring in a one-time unit?
If $\lambda$ is the parameter of an exponential distribution it's called the rate and the mean is $\frac 1{\lambda}$ which is expected time. So if the bus passes with exponential distribution having parameter $\lambda=2/\text{sec}$ then you expect a bus to pass in $\frac 12 $ second.