The number of cars that arrive at a parking lot follow a Poisson process with $\lambda > 0$. Be $T_1,T_2 ...$ The time of arrive between each car, so that $T_1 + T_2 + T_2 ... + T_n$ is the arrival time of the nth car
to which converges in probability
$\frac{1}{n} \sum_{i=1}^{n}T_{i}^2$ when $n \rightarrow \infty $
I am not sure what to do with the summation, i think that the solution has something to do with the exponential distribution
the mean squared + variance $\rightarrow \frac{1}{\lambda^2} + \frac{1}{\lambda^2} = \frac{2}{\lambda^2} $
because n tends to infinity so the the dispersion measure will tend to an exponenential distribution (SLLN)
i will appreciate if someone can clarify me more on the solution,thanks
$(T_i)$ is i.i.d and so is $(T_i^{2})$. Hence, SLLN shows that $\frac 1 n \sum\limits_{i=1}^{n} T_i^{2} \to ET_1^{2}$. This is just the second moment of the $exp(\lambda)$ distribution.