Suppose \( U \sim \text{Uniform}(0, 1) \)
Then the probability density function of \( U \), \( f_U(u) = \left\{\begin{matrix} 1, && \text{if} ~~ u \in (0, 1) \\ 0, && \text{if} ~~ u \notin (0, 1)\end{matrix} \right. \)
The cumulative distribution function of \( U \), \( F_U(u) = \Pr\{U \leq u\} = \int_{-\infty}^u f_U(w)dw \)
In general for any Borel-measurable set \( A \), \( \Pr\{U \in A\} = \int_A f_U(u)du \)
And the expectation of \( g(U) \), \( E[g(U)] = \int_0^1 g(u)f_U(u)du \)
So it is a function that completely characterize an absolutely continuous random variable.
Then the probability density function of \( U \), \( f_U(u) = \left\{\begin{matrix} 1, && \text{if} ~~ u \in (0, 1) \\ 0, && \text{if} ~~ u \notin (0, 1)\end{matrix} \right. \)
The cumulative distribution function of \( U \), \( F_U(u) = \Pr\{U \leq u\} = \int_{-\infty}^u f_U(w)dw \)
In general for any Borel-measurable set \( A \), \( \Pr\{U \in A\} = \int_A f_U(u)du \)
And the expectation of \( g(U) \), \( E[g(U)] = \int_0^1 g(u)f_U(u)du \)
So it is a function that completely characterize an absolutely continuous random variable.