Independent classes of random variables  (Page 3/6)

Remark . While it is true that every independent pair of normally distributed random variables is joint normal, not every pair of normally distributed random variables has thejoint normal distribution.

Independent classes

Since independence of random variables is independence of the events determined by the random variables, extension to general classes is simple and immediate.

Definition

A class $\{X_i:i\in J\}$ of random variables is (stochastically) independent iff the product rule holds for every finite subclass of two or more.

Remark . The index set J in the definition may be finite or infinite.

For a finite class $\{X_i:1\le i\le n\}$ , independence is equivalent to the product rule

Since we may obtain the joint distribution function for any finite subclass by letting the arguments for the others be ∞ (i.e., by taking the limits as the appropriate t _i increase without bound), the single product rule suffices to account for all finite subclasses.

Absolutely continuous random variables

If a class $\{X_i:i\in J\}$ is independent and the individual variables are absolutely continuous (i.e., have densities), then any finite subclass is jointly absolutelycontinuous and the product rule holds for the densities of such subclasses

Similarly, if each finite subclass is jointly absolutely continuous, then each individual variable is absolutely continuous and the product rule holds for the densities.Frequently we deal with independent classes in which each random variable has the same marginal distribution. Such classes are referred to as iid classes (an acronym for i ndependent, i dentically d istributed). Examples are simple random samplesfrom a given population, or the results of repetitive trials with the same distribution on the outcome of each component trial. A Bernoulli sequence is a simple example.

Simple random variables

Consider a pair $\{X,Y\}$ of simple random variables in canonical form