Independent classes of random variables (Page 3/6)

Applied probability Page 3 / 6

This image consist of three elements. There is a vertical line, a horizontal line and square comprised of four square assembled together. On each of the lines there are three points indicated by a bold dot and a number. Original numbers are represented in bold while calculated ones are in italics. Vertical line has these points labeled ascending 0.3 in bold, 0.4 in italics and 0.3 in italics. The horizontal line has points progressing to the right labeled 0.2 in bold, 0.5 in italics, and 0.3 in italics. There are points at all the intersections of the lines. — Joint and marginal probabilities from partial information.

The joint normal distribution

A pair ${X, Y}$ has the joint normal distribution iff the joint density is

f_{X Y} (t, u) = \frac{1}{2 π σ_{X} σ_{Y} {(1 - ρ^{2})}^{1 / 2}} e^{- Q (t, u) / 2}

where

Q (t, u) = \frac{1}{1 - ρ^{2}} [{(\frac{t - μ_{X}}{σ_{X}})}^{2} - 2 ρ (\frac{t - μ_{X}}{σ_{X}}) (\frac{u - μ_{Y}}{σ_{Y}}) + {(\frac{u - μ_{Y}}{σ_{Y}})}^{2}]

The marginal densities are obtained with the aid of some algebraic tricks to integrate the joint density. The result is that $X \sim N (μ_{X}, σ_{X}^{2})$ and $Y \sim N (μ_{Y}, σ_{Y}^{2})$ . If the parameter ρ is set to zero, the result is

f_{X Y} (t, u) = f_{X} (t) f_{Y} (u)

so that the pair is independent iff $ρ = 0$ . The details are left as an exercise for the interested reader.

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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.

A normal pair not joint normally distributed

We start with the distribution for a joint normal pair and derive a joint distribution for a normal pair which is not joint normal. The function

φ (t, u) = \frac{1}{2 π} exp (- \frac{t^{2}}{2} - \frac{u^{2}}{2})

is the joint normal density for an independent pair ( $ρ = 0$ ) of standardized normal random variables. Now define the joint density for a pair ${X, Y}$ by

f_{X Y} (t, u) = 2 φ (t, u) in the first and third quadrants, and zero elsewhere

Both $X \sim N (0, 1)$ and $Y \sim N (0, 1)$ . However, they cannot be joint normal, since the joint normal distribution is positive for all $(t, u)$ .

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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 ${X_{i} : 1 \leq i \leq n}$ , independence is equivalent to the product rule

F_{X_{1} X_{2} \dots X_{n}} (t_{1}, t_{2}, \dots, t_{n}) = \prod_{i = 1}^{n} F_{X_{i}} (t_{i}) for all (t_{1}, t_{2}, \dots, t_{n})

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

f_{X_{i 1} X_{i 2} \dots X_{i m}} (t_{i 1}, t_{i 2}, \dots, t_{i m}) = \prod_{k = 1}^{m} f_{X_{i k}} (t_{i k}) for all (t_{1}, t_{2}, \dots, t_{n})

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

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Source: OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6

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