Writing a r.v. in set notation

In summary, a random variable X is defined as a set of all outcomes in a sample space that map to a Borel subset of the real numbers, where F is a sigma algebra. For a Borel measurable function f, the set notation definition for f(X) is similar, where the set contains all outcomes in the sample space that map to a Borel subset of the real numbers, but now with f applied to X and sigma(X) as the sigma algebra.
  • #1
nasshi
18
0
The definition I have for a random variable is
[tex]X=\lbrace \omega \in \Omega \vert X(\omega) \in B \rbrace \in F[/tex] where F is a sigma algebra and B is a Borel subset of R.

Using function composition, how would one write a similar set notation definition for f(X), where f is a Borel measurable function?
[tex]f(X)=\lbrace \omega \in \Omega \vert f(X(\omega)) \in B \rbrace \in \sigma(X)[/tex] where [tex]\sigma(X)[/tex] is a sigma algebra and B is a Borel subset of R??
 
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  • #2
Your espression looks right. In both cases, there is an implicit assumption that X and f(X) are real-valued.
 

Related to Writing a r.v. in set notation

What is a random variable?

A random variable is a numerical value that represents the outcome of a random event or experiment. It is typically denoted by a capital letter, such as X, and can take on different values based on the probability of each outcome.

What is set notation?

Set notation is a mathematical notation used to describe sets, which are collections of objects or elements. It typically includes curly braces { } and a list of elements separated by commas, such as {1,2,3,4}.

How is a random variable written in set notation?

A random variable can be written in set notation as {X: X = x}, where X is the variable and x represents a specific outcome or value it can take on. This notation indicates that the set contains all possible values of X.

Can a random variable have more than one value in set notation?

Yes, a random variable can have multiple values in set notation. For example, if X represents the outcome of rolling a 6-sided die, the set notation could be written as {X: X = 1,2,3,4,5,6}.

How is a random variable written in set notation for a discrete probability distribution?

For a discrete probability distribution, the set notation for a random variable X is typically written as {X: X = x1, x2, x3,...xn}, where x1, x2, x3,...xn represent the possible outcomes with their respective probabilities. For example, if X represents the number of heads when flipping a coin twice, the set notation could be written as {X: X = 0,1,2} with probabilities {0.25, 0.5, 0.25}.

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