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Random variables as functions

We consider in this chapter real random variables (i.e., real-valued random variables). In the chapter "Random Vectors and Joint Distributions" , we extend the notion to vector-valued random quantites. The fundamental idea of a real random variable is the assignment of a real number to each elementary outcome ω in the basic space Ω . Such an assignment amounts to determining a function X , whose domain is Ω and whose range is a subset of the real line R . Recall that a real-valued function on a domain (say an interval I on the real line) is characterized by the assignment of a real number y to each element x (argument) in the domain. For a real-valued function of a real variable, it is often possible to write a formula or otherwise state a rule describing the assignmentof the value to each argument. Except in special cases, we cannot write a formula for a random variable X . However, random variables share some important general properties offunctions which play an essential role in determining their usefulness.

Mappings and inverse mappings

There are various ways of characterizing a function. Probably the most useful for our purposes is as a mapping from the domain Ω to the codomain R . We find the mapping diagram of Figure 1 extremely useful in visualizing the essential patterns. Randomvariable X , as a mapping from basic space Ω to the real line R , assigns to each element ω a value t = X ( ω ) . The object point ω is mapped, or carried, into the image point t . Each ω is mapped into exactly one t , although several ω may have the same image point.

A rectangle with an upper case omega in the top left corner and a lower case omega in the middle next to an arrow pointing down and to the right to a line segment situated below the rectangle. On the left hand side of the line there is an upper case R. The arrow points to a lower case t. Below the line segment there is the function: t=X(ω) A rectangle with an upper case omega in the top left corner and a lower case omega in the middle next to an arrow pointing down and to the right to a line segment situated below the rectangle. On the left hand side of the line there is an upper case R. The arrow points to a lower case t. Below the line segment there is the function: t=X(ω)
The basic mapping diagram t = X ( ω ) .

Associated with a function X as a mapping are the inverse mapping X - 1 and the inverse images it produces. Let M be a set of numbers on the real line. By the inverse image of M under the mapping X , we mean the set of all those ω Ω which are mapped into M by X (see Figure 2). If X does not take a value in M , the inverse image is the empty set (impossible event). If M includes the range of X , (the set of all possible values of X ), the inverse image is the entire basic space Ω . Formally we write

X - 1 ( M ) = { ω : X ( ω ) M }

Now we assume the set X - 1 ( M ) , a subset of Ω , is an event for each M . A detailed examination of that assertion is a topic in measure theory . Fortunately, the results of measure theory ensure that we may make the assumption for any X and any subset M of the real line likely to be encountered in practice. The set X - 1 ( M ) is the event that X takes a value in M . As an event, it may be assigned a probability.

Diagram with a rectangle in the background. On the upper left-hand corner of the rectangle there is an upper-case omega and in the center there is a shaded oval containing an upper-case E. Line segments extend downwards from the oval to a shaded rectangle, split down the middle horizontally by another line segment. In the middle of the shaded rectangle there is an upper-case M. Diagram with a rectangle in the background. On the upper left-hand corner of the rectangle there is an upper-case omega and in the center there is a shaded oval containing an upper-case E. Line segments extend downwards from the oval to a shaded rectangle, split down the middle horizontally by another line segment. In the middle of the shaded rectangle there is an upper-case M.
E is the inverse image X - 1 ( M ) .

Some illustrative examples.

  1. X = I E where E is an event with probability p . Now X takes on only two values, 0 and 1. The event that X take on the value 1 is the set
    { ω : X ( ω ) = 1 } = X - 1 ( { 1 } ) = E
    so that P ( { ω : X ( ω ) = 1 } ) = p . This rather ungainly notation is shortened to P ( X = 1 ) = p . Similarly, P ( X = 0 ) = 1 - p . Consider any set M . If neither 1 nor 0 is in M , then X - 1 ( M ) = If 0 is in M , but 1 is not, then X - 1 ( M ) = E c If 1 is in M , but 0 is not, then X - 1 ( M ) = E If both 1 and 0 are in M , then X - 1 ( M ) = Ω In this case the class of all events X - 1 ( M ) consists of event E , its complement E c , the impossible event , and the sure event Ω .
  2. Consider a sequence of n Bernoulli trials, with probability p of success. Let S n be the random variable whose value is the number of successes in the sequence of n component trials. Then, according to the analysis in the section "Bernoulli Trials and the Binomial Distribution"
    P ( S n = k ) = C ( n , k ) p k ( 1 - p ) n - k 0 k n
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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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