R: 주어진 예측변수 집합에 대해 가능한 모든 선형 회귀 모델 계산

**Introduction** Computing the probability distribution of a variable is a fundamental task in statistics and data science. Depending on the context—whether the underlying mechanism is known, data are available, or the goal is to make inference—different techniques are employed. The following methods are the most widely used. | Method | Short Summary | |--------|---------------| | **Analytical derivation** | When the random variable is defined by a mathematical model (e.g., a sum of independent Bernoulli trials), the distribution can be derived directly from the definition using probability rules, leading to known forms such as the binomial, Poisson, or normal distributions. | | **Empirical frequency** | By collecting observations of the variable, one can tabulate how often each outcome occurs. The relative frequencies form an empirical distribution that approximates the true distribution, especially with large samples. | | **Parametric estimation** | If the variable is believed to follow a particular family of distributions (normal, exponential, etc.), its parameters can be estimated from data via methods such as maximum likelihood estimation (MLE) or the method of moments. The resulting parameterized density or mass function then represents the distribution. | | **Bayesian inference** | Treats distribution parameters as random and updates a prior distribution to a posterior using observed data. The posterior distribution gives a full probabilistic description of the variable, capturing uncertainty about the parameters. | | **Simulation/Monte Carlo** | Generates synthetic data according to a specified model or random process. By repeating the simulation many times, the resulting histogram of simulated outcomes approximates the probability distribution, useful when analytical solutions are intractable. | | **Kernel density estimation (KDE)** | A non‑parametric approach that smooths observed data points with a kernel function to estimate a continuous probability density. KDE is effective for capturing complex shapes without assuming a parametric family. | | **Transformation methods** | For variables derived from others via deterministic transformations (e.g., \(Y = g(X)\)), the distribution of \(Y\) can be found by applying the transformation theorem or Jacobian methods to the distribution of \(X\). | | **Moment generating functions (MGFs)** | When MGFs or characteristic functions are known, they can be inverted to recover the probability distribution. This is often used for theoretical derivations or to confirm distributional properties. | **Conclusion** The choice of method depends on whether the distribution is known analytically, must be inferred from data, or needs to be approximated via simulation or non‑parametric techniques. In practice, a combination of empirical estimation, parametric fitting, and simulation often yields a robust understanding of a variable’s probability distribution.