Jointly Distributed Random Variables
Stub. Joint cumulative distribution functions, joint probability mass functions, joint probability density functions, and worked examples combining them; independence of two random variables and how to recognize it from a joint pdf; the distribution of a sum of independent random variables (convolution, discrete and continuous); transformations of a joint pdf under linear, one-to-one, and many-to-one mappings; and a pointer to the joint Gaussian distribution, covered alongside MMSE estimation.
Extends the single-variable machinery (CDF, pmf, pdf) to pairs (and tuples) of random variables observed together: joint CDFs as the universal object, joint pmfs/pdfs when they exist, and marginals recovered by summing or integrating out the other variable. Independence for random variables generalizes independence of events (conditional-probability-and-independence) to the statement that the joint distribution factors into the marginals, and the sum of independent random variables is computed by convolution — the machinery the law of large numbers and CLT (concentration-inequalities-and-limit-theorems) operate on. Also covers transforming a joint pdf under a mapping (linear, one-to-one, or many-to-one), the multivariate generalization of transforming a single random variable (continuous-random-variables-and-distributions). The joint Gaussian distribution is not covered here — see the note below.
Outline (TODO — flesh out each)
NoteTODO — all sections below
This whole note is still a stub. Needs Ross chapter 6 (jointly
distributed random variables) and the corresponding Hajek material.
Source text not yet extracted into sources/ — do not write from memory.
- Joint cumulative distribution functions
- Joint probability mass functions
- Joint probability density functions — marginals
- Additional examples using joint distributions
- Definition of independence for two random variables
- Determining from a pdf whether independence holds
- Sums of integer-valued random variables — convolution of pmfs
- Sums of jointly continuous-type random variables — convolution of pdfs
- Transformations of jointly distributed random variables — linear, one-to-one (Jacobian), and many-to-one mappings
NoteThe joint Gaussian distribution lives in a companion note
The bivariate/joint Gaussian distribution is covered in expectation-covariance-and-mmse-estimation (still a stub as of this writing — see its own TODO), grouped with MMSE estimation because that is where the linear MMSE estimator is shown to coincide with the true MMSE estimator for jointly Gaussian variables.