There is a random quantity $L$, which is evenly distributed from 0 to $l_{\max}$.
How to find the dependence of the probability that the variable $x$ (positive) will be less than the value of the random variable $L$?
Intuitively, this will be an exponential dependence from zero to $l_{\max}$.
But I cannot prove it to myself. Any thoughts in which direction to think?
Best Answer
The random quantity L has the following distribution functio:
$$ F_{L}(x)=\begin{cases} 0, & x<0\\ \frac{x}{l_{max}}, & 0\leq x<l_{max}\\ 1, & x\geq t_{stay} \end{cases} $$
The distribution function has the meaning of the probability that the value of a random variable l will be less than an arbitrary number x.
$$ F_{L}(x)=P(l<x). $$
$l < x$ event, forms a complete group with $l\geq x$ event. Then the probability of this event is found by the formula:
$$ P(x\leq l)=1-P(l<x). $$
Then the required probability is found by the formula:
$$ p(x)=1-\begin{cases} 0, & x<0\\ \frac{x}{l_{max}}, & 0\leq x<l_{max}\\ 1, & x\geq l_{max} \end{cases} $$