Random variables and stochastic processes involve the assignment of numerical values to experimental outcomes. A random variable is a function that maps outcomes of an experiment to numerical values. Random variables can be discrete (DV) or continuous (CV). Discrete variables have distinct values, while continuous variables can take any value in a range. The distribution function of a random variable X is the probability that X is less than or equal to a given value. It is a monotonic function, and its derivative is the probability density function (pdf). The pdf describes the likelihood of the variable taking on a specific value. For discrete variables, the pdf is the probability mass function, while for continuous variables, it is the density. The expectation (mean) of a random variable is the average value over many trials. The first moment is the expected value, and the second moment is the mean-squared value. The variance is the second central moment, representing the spread of the distribution. The standard deviation is the square root of the variance. For complex-valued variables, absolute moments are useful. The nth absolute moment is the expected value of the absolute value of the variable raised to the nth power. Joint probability density functions describe the distribution of two or more random variables. If two variables are independent, their joint pdf is the product of their individual pdfs. Independence implies zero covariance, but zero covariance does not necessarily imply independence. Covariance measures the linear relationship between two variables. If two variables are independent, their covariance is zero. However, zero covariance does not guarantee independence. Uncorrelated variables have zero covariance, but this does not imply orthogonality. The variance of a sum of random variables includes the variances of each variable and twice the covariance between them. For a linear combination of variables, the expectation and variance can be calculated based on the coefficients and the variances and covariances of the individual variables.Random variables and stochastic processes involve the assignment of numerical values to experimental outcomes. A random variable is a function that maps outcomes of an experiment to numerical values. Random variables can be discrete (DV) or continuous (CV). Discrete variables have distinct values, while continuous variables can take any value in a range. The distribution function of a random variable X is the probability that X is less than or equal to a given value. It is a monotonic function, and its derivative is the probability density function (pdf). The pdf describes the likelihood of the variable taking on a specific value. For discrete variables, the pdf is the probability mass function, while for continuous variables, it is the density. The expectation (mean) of a random variable is the average value over many trials. The first moment is the expected value, and the second moment is the mean-squared value. The variance is the second central moment, representing the spread of the distribution. The standard deviation is the square root of the variance. For complex-valued variables, absolute moments are useful. The nth absolute moment is the expected value of the absolute value of the variable raised to the nth power. Joint probability density functions describe the distribution of two or more random variables. If two variables are independent, their joint pdf is the product of their individual pdfs. Independence implies zero covariance, but zero covariance does not necessarily imply independence. Covariance measures the linear relationship between two variables. If two variables are independent, their covariance is zero. However, zero covariance does not guarantee independence. Uncorrelated variables have zero covariance, but this does not imply orthogonality. The variance of a sum of random variables includes the variances of each variable and twice the covariance between them. For a linear combination of variables, the expectation and variance can be calculated based on the coefficients and the variances and covariances of the individual variables.