Relationship between truncated random distribution and original distribution

The probability density function of the known random distribution is $f_X(x)$, The domain of definition is $D$. Now its definition domain is intercepted as $E$, among $E \subseteq D$, That is, take random variables according to the distribution until the value of the variable falls in $E$ in . The probability density function of the distribution of the intercepted random variables and $f_X(x)$ What is the relationship ？

Answer that question , First, let the probability density function after interception be $f_U(x)$, set up $a=\min{E}$（ If $E$ No lower bound , Make $a$ Express $-\infty$）.$\forall x \in E$：

$$\begin{aligned} \int_a^x{f_U(t)dt} &= \int_a^x{f_X(t)dt} + \left(1 - \int_E{f_X(t)dt}\right)\int_a^x{f_X(t)dt} + \cdots\\ \int_a^x{f_U(t)dt} &= \sum_{n=1}^\infty{\left(1-\int_E{f_X(t)dt}\right)}^n \int_a^x{f_X(t)dt}\\ \int_a^x{f_U(t)dt} &= \left(\int_E{f_X(t)dt}\right)^{-1} \int_a^x{f_X(t)dt}\\ {d \over dx}\int_a^x{f_U(t)dt} &= \left(\int_E{f_X(t)dt}\right)^{-1} {d \over dx}\int_a^x{f_X(t)dt}\\ f_U(x) &= \left(\int_E{f_X(t)dt}\right)^{-1} f_X(x) \end{aligned}$$

So the random distribution will not change in shape , But it will get higher .