# probability density function

#### sussan

##### New Member
hi, can any one let me know with example "probability density function". I understand in more details about PDF with exaple. please help me

#### BGM

##### TS Contributor
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.