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Often we have more than one random variable. Each can be considered separately, but usually they have some probabilistic ties which must be taken into account when they are considered jointly. We treat the joint case by considering the individual random variables as coordinates of a random vector. We extend the techniques for a single random variable to the multidimensional case. To simplify exposition and to keep calculations manageable, we consider a pair of random variables as coordinates of a two-dimensional random vector. The concepts and results extend directly to any finite number of random variables considered jointly. If the joint distribution for a random vector is known, then the distribution for each of the component random variables may be determined. These are known as marginal distributions. In general, the converse is not true. However, if the component random variables form an independent pair, the treatment in that case shows that the marginals determine the joint distribution.

Introduction

A single, real-valued random variable is a function (mapping) from the basic space Ω to the real line. That is, to each possible outcome ω of an experiment there corresponds a real value t = X ( ω ) . The mapping induces a probability mass distribution on the real line, which provides a means of making probabilitycalculations. The distribution is described by a distribution function F X . In the absolutely continuous case, with no point mass concentrations, the distribution may also bedescribed by a probability density function f X . The probability density is the linear density of the probability mass along the real line (i.e., mass per unit length).The density is thus the derivative of the distribution function. For a simple random variable, the probability distribution consists of a point mass p i at each possible value t i of the random variable. Various m-procedures and m-functions aid calculations for simple distributions. In the absolutely continuous case, a simple approximationmay be set up, so that calculations for the random variable are approximated by calculations on this simple distribution.

Often we have more than one random variable. Each can be considered separately, but usually they have some probabilistic ties which must be taken into account when theyare considered jointly. We treat the joint case by considering the individual random variables as coordinates of a random vector . We extend the techniques for a single random variable to the multidimensional case. To simplify exposition and to keep calculationsmanageable, we consider a pair of random variables as coordinates of a two-dimensional random vector. The concepts and results extend directly to any finite number of randomvariables considered jointly.

Random variables considered jointly; random vectors

As a starting point, consider a simple example in which the probabilistic interaction between two random quantities is evident.

A selection problem

Two campus jobs are open. Two juniors and three seniors apply. They seem equally qualified, so it is decided to select them by chance. Each combination of two isequally likely. Let X be the number of juniors selected (possible values 0, 1, 2) and Y be the number of seniors selected (possible values 0, 1, 2). However there are only three possible pairs of values for ( X , Y ) : ( 0 , 2 ) , ( 1 , 1 ) , or ( 2 , 0 ) . Others have zero probability, since they are impossible. Determine the probability for each of thepossible pairs.

Solution

There are C ( 5 , 2 ) = 10 equally likely pairs. Only one pair can be both juniors. Six pairs can be one of each. There are C ( 3 , 2 ) = 3 ways to select pairs of seniors. Thus

P ( X = 0 , Y = 2 ) = 3 / 10 , P ( X = 1 , Y = 1 ) = 6 / 10 , P ( X = 2 , Y = 0 ) = 1 / 10

These probabilities add to one, as they must, since this exhausts the mutually exclusive possibilities. The probability of any other combination must be zero. We also have thedistributions for the random variables conisidered individually.

X = [ 0 1 2 ] P X = [ 3 / 10 6 / 10 1 / 10 ] Y = [ 0 1 2 ] P Y = [ 1 / 10 6 / 10 3 / 10 ]

We thus have a joint distribution and two individual or marginal distributions .

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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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