1. Cumulative distribution functions and expected values

2. Cumulative distribution function of a continuous random variable
Whereas for a discrete random variable we sum probabilities up to a point x to obtain the cumulative distribution function , in the continuous case we integrate.

3. Definition and properties of the cumulative distribution function
The CDF of a continuous random variable with density function f(x) is
The function F(x) takes values in [0,1], i.e.
, and is non-decreasing.

4. Examples
For X with pdf,
the CDF is given by

5. Angles corresponding to an imperfection on a tire
For this example,
In general, for a uniform rv on [A,B],

6. Using F(x) to compute probabilities
Let X be a continuous random variable with pdf f(x) and cdf F(X). Then for any number a,
and for any two numbers a and b,

7. Obtaining f(x) from F(x)
If X is a continuous rv with pdf f(x) and cdf F(x), then at every x at which the
derivative exists,
For ,

8. Percentiles of a continuous distributon
Let 0<p<1. The (100p)th percentile of a rv, denoted by , is defined by
The median is the 50th percentile. The text denotes the median by

9. Median of a symmetric distribution
A continuous distribution whose pdf is symmetric has median equal to the point of symmetry.

10. Expected values of continuous random variables
Expected values of continuous random variables are computed in a manner similar to the discrete case, with integrals replacing sums.

11. Definition
The expected value or mean of a continuous rv X with pdf f(x) is computed as
For a function h(X),

12. Definition of variance
The variance of a continuous random variable with pdf f(x) and mean value is
The variance may also be computed using