How to find the probability that the variable will be less than the value of the random variable?

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?

• $P(x<l)=1-\frac x {l_{max}}$ for $0 <x <l_{\max}$. Jan 29 '21 at 8:44
• thanks for the answer, but why? Jan 29 '21 at 8:51

The random quantity L has the following distribution functio:

$$F_{L}(x)=\begin{cases} 0, & x<0\\ \frac{x}{l_{max}}, & 0\leq x

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

$$l < x$$ 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

Then the required probability is found by the formula:

$$p(x)=1-\begin{cases} 0, & x<0\\ \frac{x}{l_{max}}, & 0\leq x