The concept of random vector is a multidimensional generalization of the concept of random variable.
Suppose that we conduct a probabilistic experiment and that the possible outcomes of the experiment are described by a sample space . A random vector is a vector whose value depends on the outcome of the experiment, as stated by the following definition.
Definition Let be a sample space. A random vector is a function from the sample space to the set of -dimensional real vectors :
In rigorous probability theory, the function is also required to be measurable (a concept found in measure theory - see a more rigorous definition of random vector).
The real vector associated to a sample point is called a realization of the random vector. The set of all possible realizations is called support and is denoted by .
Denote by the probability of an event . When dealing with random vectors, the following conventions are used:
If , we often write with the meaning
If , we sometimes use the notation with the meaningIn applied work, it is very commonplace to build statistical models where a random vector is defined by directly specifying , omitting the specification of the sample space altogether.
We often write instead of , omitting the dependence on .
The following example shows how a random vector can be defined on a sample space.
Example Two coins are tossed. The possible outcomes of each toss can be either tail () or head (). The sample space isThe four possible outcomes are assigned equal probabilities:If tail () is the outcome, we win one dollar, if head () is the outcome we lose one dollar. A 2-dimensional random vector indicates the amount we win (or lose) on each toss:The probability of winning one dollar on both tosses isThe probability of losing one dollar on the second toss is
This section and the next one deal with discrete random vectors and absolutely continuous random vectors, two kinds of random vectors that have special properties and are often found in applications.
Discrete random vectors are defined as follows.
Definition A random vector is discrete if
its support is a countable set;
there is a function , called the joint probability mass function (or joint pmf, or joint probability function) of , such that, for any :
The following notations are used interchangeably to indicate the joint probability mass function:In the second and third notation the components of the random vector are explicitly indicated.
Example Suppose is a -dimensional random vector whose components ( and ) can take only two values: or . Furthermore, the four possible combinations of and are all equally likely. is an example of a discrete random vector. Its support is Its probability mass function is
Absolutely continuous random vectors are defined as follows.
The following notations are used interchangeably to indicate the joint probability density function:In the second and third notation the components of the random vector are explicitly indicated.
Example Suppose is a -dimensional random variable whose components ( and ) are independent uniform random variables (on the interval ). is an example of an absolutely continuous random variable. Its support isIts joint probability density function isThe probability that the realization of falls in the rectangle is
Random vectors, also those that are neither discrete nor absolutely continuous, are often described using their joint distribution function.
Definition Let be a random vector. The joint distribution function (or joint df, or joint cumulative distribution function, or joint cdf) of is a function such thatwhere the components of and are denoted by and respectively, for .
The following notations are used interchangeably to indicate the joint distribution function:In the second and third notation the components of the random vector are explicitly indicated.
Sometimes, we talk about the joint distribution of a random vector, without specifying whether we are referring to the joint distribution function, or to the joint probability mass function (in the case of discrete random vectors), or to the joint probability density function (in the case of absolutely continuous random vectors). This ambiguity is legitimate, since
the joint probability mass function completely determines (and is completely determined by) the joint distribution function of a discrete random vector;
the joint probability density function completely determines (and is completely determined by) the joint distribution function of an absolutely continuous random vector.
In the remainder of this lecture, we use the term joint distribution when we are making statements that apply both to the distribution function and to the probability mass (or density) function of a random vector.
The following subsections contain more details about random vectors.
A random matrix is a matrix whose entries are random variables. It is not necessary to develop a separate theory for random matrices, because a random matrix can always be written as a random vector. Given a random matrix , its vectorization, denoted by , is the random vector obtained by stacking the columns of on top of each other.
Example Let be the following random matrix:The vectorization of is the following random vector:
When is a discrete random vector, then we say that is a discrete random matrix and the joint probability mass function of is just the joint probability mass function of . By the same token, when is an absolutely continuous random vector, then we say that is an absolutely continuous random matrix and the joint probability density function of is just the joint probability density function of .
Let be the -th component of a -dimensional random vector . The distribution function of is called marginal distribution function of . If is discrete, then is a discrete random variable and its probability mass function is called marginal probability mass function of . If is absolutely continuous, then is an absolutely continuous random variable and its probability density function is called marginal probability density function of .
The process of deriving the distribution of a component of a random vector from the joint distribution of is known as marginalization. Marginalization can also have a broader meaning: it can refer to the act of deriving the joint distribution of a subset of the set of components of from the joint distribution of . For example, if is a random vector having three components (, and ), we can marginalize the joint distribution of , and to find the joint distribution of and (in this case we say that is marginalized out of the joint distribution of , and ).
Let be the -th component of a -dimensional discrete random vector . The marginal probability mass function of can be derived from the joint probability mass function of as follows:where the sum is over the setIn other words, the probability that is obtained as the sum of the probabilities of all the vectors in such that their -th component is equal to .
Let be the -th component of a discrete random vector . Marginalizing out of the joint distribution of , we can obtain the joint distribution of the remaining components of , i.e. we can obtain the joint distribution of the random vector defined as follows:The joint probability mass function of is computed as follows:where the sum is over the setIn other words, the joint probability mass function of can be computed by summing the joint probability mass function of over all values of that belong to the support of .
Let be the -th component of a -dimensional absolutely continuous random vector . The marginal probability density function of can be derived from the joint probability density function of as follows:In other words, the joint probability density function, evaluated at , is integrated with respect to all variables except (so it is integrated a total of times).
Let be the -th component of an absolutely continuous random vector . Marginalizing out of the joint distribution of , we can obtain the joint distribution of the remaining components of , i.e. we can obtain the joint distribution of the random vector defined as follows:The joint probability density function of is computed as follows:In other words, the joint probability density function of can be computed by integrating the joint probability density function of with respect to .
Note that, if is absolutely continuous, thenHence, by taking the -th order cross-partial derivative with respect to of both sides of the above equation, we obtain
We report here a more rigorous definition of random vector.
Definition Let be a probability space. Let be a function . Let be the Borel -algebra of (i.e., the smallest -algebra containing all open hyper-rectangles in ). If for any , then is a random vector on .
Thus, if satisfies this property, we are allowed to define because the set is measurable by the very definition of random vector.
Some solved exercises on random vectors can be found below.
Let be a discrete random vector and denote its components by and . Let the support of be the set of all vectors such that their entries belong to the set of the first three natural numbers, i.e., whereLet the joint probability mass function of beFind .
Trivially, we need to evaluate the joint probability mass function at the point , i.e.,
Let be a discrete random vector and denote its components by and . Let the support of be the set of all vectors such that their entries belong to the set of the first three natural numbers, i.e.,whereLet the joint probability mass function of beFind .
There are only two possible cases that give rise to the occurrence . These cases areandTherefore, since these two cases are disjoint events, we can use the additivity of probability:
Let be a discrete random vector and denote its components by and . Let the support of beand its joint probability mass function beDerive the marginal probability mass functions of and .
The support of is
We need to compute the probability of each element of the support of :
Thus, the probability mass function of is
The support of is
We need to compute the probability of each element of the support of :
Thus, the probability mass function of is
Let be a absolutely continuous random vector and denote its components by and . Let the support of be i.e., the set of all vectors such that the first component belongs to the interval and the second component belongs to the interval . Let the joint probability density function of beCompute .
By the very definition of joint probability density function:
Let be a absolutely continuous random vector and denote its components by and . Let the support of be i.e., the set of all vectors such that the first component belongs to the interval and the second component belongs to the interval . Let the joint probability density function of beCompute .
First of all note that if and only if . Using the definition of joint probability density function, we obtain
Now, note that, when , the inner integral isTherefore,
Let be a absolutely continuous random vector and denote its components by and . Let the support of be (i.e., the set of all -dimensional vectors with positive entries) and its joint probability density function beDerive the marginal probability density functions of and .
The support of is(recall that and )
We can find the marginal density by integrating the joint density with respect to :
When
,
then
and the above integral is trivially equal to
.
Thus, when
,
then
.
When , thenbut the first of the two integrals is zero since when ; as a consequence,So, putting pieces together, the marginal density function of is
Obviously, by symmetry, the marginal density function of is
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