The time between release from prison and the commission of another crime is uniformly distributed between 0 and 5 years for a high-risk group. Give the equation and graph the pdf for X, the time between release and the commission of another crime for this group. What percent of this group will commit another crime within two years of their release from prison?

Solution

Let $X$ denote the time between release from prison and the commission of another crime. $X\sim U(0, 5)$.

The probability density curve is
$$ \begin{aligned} f(x)=\left\{ \begin{array}{ll} \frac{1}{5 - 0}=0.2 , & \hbox{$0 \leq x\leq 5$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{aligned} $$

The graph of the pdf $f(x)$ is

uniform distribution
uniform distribution

The probability that the group will commit another crime within two years of their release from prison is

$$ \begin{aligned} P(X\leq 2) &= \int_{0}^{2} f(x) \; dx\\ &= \frac{1}{5-0}\int_{0}^{2} \; dx\\ &= 0.2\big[x\big]_{0}^{2} \\ &= 0.2\big[ 2- 0]\\ &= 0.4 \end{aligned} $$

Thus $40$ percent of this group will commit another crime within two years of their release from prison.

Further Reading