finding X for density function

A city's fire brigade is deployed approximately every 2 days to extinguish a fire. The number of operations X per week is assumed to be Poisson distributed. Calculate the density and the distribution function of the time T that elapses between two fires. Represent both functions graphically.

I have this formula for the density function:

$f(x)=\frac{\mu^x}{x!} \cdot \exp^{-\mu}$

$\mu = 2$ because $2$ days pass between fires I think but I am not sure what $X$ should be.

Solution 1:

If the model il poisson with a mean of $2$ days, this means that $T$, the interarrival time between two fires, is exponentially distributed with mean $1/2$ days, say

$$f_T(t)=2 e^{-2t}~[t>0]$$

the proof is very simple. Understood this, the rest is immediate