Conditional expectation for dependent variables

Question

The expected number of potholes on a road depends on the time since the last repair. The expected number is given by $ct^2$ in function of the time $t$ since the last repair. The constant is $c = 3.2 $ potholes per year. The repairs happen every 5 years (exactly). Calculate the expected value and the variance of the number of potholes if the time since the last repair is unknown.

What I've done

I computed some formulas but I'm stuck with the next steps, any idea?

$$E[X \vert T=t]=ct^2$$

$$E[X \vert T=t]=\int x\frac{f_{X,T}(x,t)}{f_T(t)}dx$$

$$E[X]=E[E[X \vert T]]=E[ct^2]=cE[T^2]=c(E[T]^2+Var[T])$$

$$Var[X]=E[Var[X \vert T]]+Var[E[X \vert T]]$$

Answer 1

I can't comment since I don't have the appropriate number of reputation points, but here it goes.

First, is there any assumption on whether the random variable $T$ is? Can $T$ be appropriately assumed to be monotonically decreasing? Without any assumption on $T$, this cannot be done. Also, is $T$ continuous?

Also, for reference,

$$ E[X] = \int_{-\infty}^{\infty} E[X|T=t] P(T=t) $$

I think there is probably more information missing on the distribution of the random variable $T$.

If this is not given, then perhaps a common random variable that describes stochastic processes for time intervals can be used. Since one piece of information - that the repair happens every 5 years - is known, perhaps a uniform random variable can be used, assuming that the last time a repair was conducted is unknown.