Suppose that the estimated number of customers at a bank is $\lambda$ per unit time. If $X$ is a random variable of the number of customers per unit time, then $X$ has a distribution $\text{Poisson}(\lambda)$. Then, $\mathbb E(X) = \lambda$ and $\text{Var}(X) = \lambda$.
Conversely, if $Y$ is a random variable that measures wait time between two customers, then $Y$ has a distribution $\text{exp}(\lambda)$. Then, $\mathbb E(Y) = \frac{1}{\lambda}$ and $\text{Var}(Y) = \frac{1}{\lambda^2}$.
Of course, $\text{Var}(Y)$ can be calculated easily, but is there an intuitive reason that $\text{Var}(Y) = \frac{1}{\lambda^2}$ in relation to $\text{Var}(X) = \lambda$? Is it just a coincidence that $\text{Var}(Y) = \frac{1}{\text{Var}(X)^2}$?