Zettelkasten/Fleeting Notes/Class/Bayesian Signal Processing/continuous-mixed-random-variables.md

Continuous and Mixed Random Variables

Source: probabilitycourse.com → Hossein Pishro-Nik
Date: Monday January 26th

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What is a continuous random variable?

A random variable X is continuous if its cumulative distribution function F_X(x) is continuous for x \in \mathbb{R}.

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Probability Density Functions

Continuous variables lend themselves to Probability Density Functions (PDFs). A PDF is defined as:

f_X(x) = \frac{dF_X(x)}{dx}

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Example: Uniform Distribution

Consider a uniform distribution of x between [a,b]:

CDF:

F_X = \begin{cases} 0, & x < a \\ \frac{x-a}{b-a}, & a \leq x \leq b \\ 1, & x > b \end{cases}

PDF:

f_X = \begin{cases} 0, & x < a \\ \frac{1}{b-a}, & a \leq x \leq b \\ 0, & x > b \end{cases}

Note: F_X is a continuous S-curve from 0 to 1 between a and b; f_X is a rectangular function with height \frac{1}{b-a} between a and b.

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Probability from PDF

Now that we've got some machinery, we can define:

P(x \in [a,b]) = \int_a^b f_X(x) \, dx