A random variable assigns a numerical value to each outcome in a sample space, transforming random experiments into quantities we can analyze. Random variables are either discrete, taking a finite or countable set of values, or continuous, taking any value in an interval. For discrete variables, the probability mass function (PMF) gives \(P(X = x)\) for each possible x; these probabilities must be non-negative and sum to one. For continuous variables, the probability density function (PDF) describes the distribution, and probabilities are obtained as areas under the curve: \(P(a \leq X \leq b) = \int_a^b f(x) \, dx\). Although a PDF can exceed one at a single point, the total area under the curve must equal one. Both types are unified by the cumulative distribution function (CDF), \(F(x) = P(X \leq x)\), which is non-decreasing and ranges from zero to one.
The expected value \(E(X)\) represents the long-run average of a random variable, also called the population mean \(\mu\). For discrete distributions, \(E(X) = \sum x_i P(x_i)\), and for continuous distributions, \(E(X) = \int x f(x) \, dx\). Variance quantifies spread around the mean through the identity \(\text{Var}(X) = E(X^2) - [E(X)]^2\), and important properties include \(\text{Var}(aX + b) = a^2 \text{Var}(X)\) and, for independent variables, \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\). Variance is always non-negative and equals zero only when X is constant.
Several named distributions appear repeatedly in applications. The binomial distribution models the number of successes in n independent trials, each with the same success probability p, and has PMF \(P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\) with mean \(np\) and variance \(np(1 - p)\). A binomial experiment requires a fixed number of trials, two outcomes per trial, constant probability of success, and independence. The Poisson distribution models the count of events in a fixed interval when events occur independently and at a constant average rate \(\lambda\), with PMF \(P(X = k) = \lambda^k e^{-\lambda} / k!\) and both mean and variance equal to \(\lambda\). When n is large and p is small (typically \(n \geq 20\) and \(p \leq 0.05\)), the Poisson with \(\lambda = np\) provides an excellent approximation to the binomial. The geometric distribution gives the number of trials until the first success, with PMF \(P(X = k) = (1 - p)^{k-1} p\), mean \(1/p\), variance \((1 - p) / p^2\), and the memoryless property.
For continuous settings, the uniform distribution assigns equal density to all values in \([a, b]\) with PDF \(f(x) = 1 / (b - a)\), mean \((a + b) / 2\), and variance \((b - a)^2 / 12\). The exponential distribution models waiting times between events in a Poisson process, with PDF \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), mean \(1/\lambda\), variance \(1/\lambda^2\), and the memoryless property \(P(X > s + t \mid X > s) = P(X > t)\). The normal (Gaussian) distribution is the most important continuous distribution, with PDF \(f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-(x - \mu)^2 / (2\sigma^2)}\) and the empirical rule that approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. The standard normal distribution, with mean zero and standard deviation one, serves as a reference: any normal variable can be standardized using \(Z = (X - \mu) / \sigma\), after which probabilities are read from a z-table or computed by software.