Random Variables
Random variables are measurable functions from a sample space to the real numbers. They encode uncertainty in quantitative form and are the building blocks of statistical models.
Key Points
- A random variable induces a probability distribution on the real line.
- Discrete variables are described by probability mass functions; continuous by densities.
- Transformations of random variables require careful handling of the measure.
Formulas
Cumulative distribution function
$$F_X(x) = P(X \le x)$$
Probability density function
$$F_X(x) = \int_{-\infty}^x f_X(t) \, dt$$
Expectation
$$\mathbb{E}[X] = \int x \, dF_X(x)$$
Code Example
import numpy as np
X = np.random.randn(10000)
print(X.mean(), X.var())