Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

3.8 CONTINUOUS RANDOM VARIABLES

We summarize several continuous random variables by providing expressions for FX(x) and FX(x), and describing important properties. For some continuous random variables, the cdf cannot be expressed in a closed form, including the well-known Gaussian random variable. Regarding distribution parameters, we use the following notation for consistency:

• μ for the mean (a location parameter).
• for the variance (a scale parameter).
• σ for the standard deviation (a scale parameter).
• c for a general location parameter (other than the mean).
• α for a general scale parameter (other than the standard deviation).
• λ for a general location/scale parameter.
• r for a general shape parameter; also for degrees of freedom of Student's t distribution.
• α and β together are the location, scale, and shape parameters for the beta distribution.
• m and n for the degrees of freedom for the F-distribution.
• N for the degrees of freedom of the chi and chi-square distributions.

Location is an indication of the “center” of the pdf (it could be exactly in the middle depending on the distribution family), and scale refers to a measure of its “width.” Specific definitions of these types of parameters are provided in Chapter 5. There is no definite pattern to the shape of a distribution when r is varied, as some pdfs can have a wide range of shapes. Example plots of the pdfs are included ...