# 4.4 Random Variables, Density Functions

## PDF To CDF

Let $$X$$ be a uniformly distributed random variable on the interval [1,12]. What is the value of the Cumulative Distribution Function (CDF) at 8? Please enter your answer as a decimal with two significant figures.

Exercise 1

Since $$X$$ has a uniform distribution, $$PDF_X(x)=\frac{1}{12}$$ for $$x\in [1, 12]$$ and 0 otherwise. $$CDF_X(x)=\Pr[X\leq x]=\sum_{k=1}^xPDF_X(k)$$. Plugging in $$x=8$$, we get $$CDF_X(8)=\sum_{k=1}^8\frac{1}{12}=\frac{8}{12}=\frac{2}{3}$$.