Probability and Random Variables

• Random Variables
  • Cumulative distribution function
  • Probability density function
  • Expectation
  • Characteristic function
  • Moment generating function
  • Second characteristic function
  • Cumulant generating function
• Gaussian distribution
  • Standard normal distribution
  • Error function and its properties
  • One sided Gaussian distribution
  • Truncated normal distribution
• Basic inequalities
  • Markov’s inequality
  • Chebyshev’s inequality
  • Boole’s inequality
  • Fano’s inequality
  • Cramér–Rao inequality
  • Hoeffding’s inequality
  • Jensen’s inequality
  • Bernstein inequalities
  • Chernoff’s inequality
  • Fréchet inequalities
• Two variables
  • Conditional density
  • Independent variables
• Expectation
  • Independent variables
  • Uncorrelated variables
• Complex random variable
• Random vectors
• Gaussian random vector
  • Whitening