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A random variable X: Ω → X(Ω) X: Ω → X (Ω) is said to be discrete, when there is a finite or countable set of values Y ⊆ X(Ω) Y ⊆ X (Ω) such that P(X ∈ Y) = 1 P (X ∈ Y) = 1. The function p: Y → [0, 1] p: Y → [0, 1] defined by. p(x) = P(X = x) p (x) = P (X = x) , for all x ∈ Y x ∈ Y. is called the probability mass ...
Random variable. A random variable X is defined as a map X: Ω → R such that, for any x ∈ R, the set {ω ∈ Ω ∣ X(ω) ≤ x} is an element of A, ergo, an element of Pr 's domain to which a probability can be assigned. We can think of X as a "realisation" of Ω, in that it assigns a real number to each outcome in Ω.
By the way, this definition should more accurately be called "finite range", since it does not cover some extremely famous examples of discrete random variables like a geometric random variable, for instance.
There are various definitions of "discrete random variable" that are used: Let (Ω, F, P) be a probability space (the sample space Ω can be uncountably infinite). Definition 1: A random variable X: Ω → R is said to be discrete if the image X(Ω) is a finite or countably infinite set. Definition 2 (not equivalent): A random variable X: Ω ...
Here is the defintion of discrete random variable from "An introduction to probability and statistics" by Rohatgi. Let (Ω, S, P) (Ω, S, P) be a probability space. An random variable X X defined on this space is said to be discrete if there exists a countable set E ⊂R E ⊂ R such that P{X ∈ E} = 1 P {X ∈ E} = 1.
2. I have two different definitions defining discrete random variables (wikipedia and a textbook) Def 1 : X: Ω → R X: Ω → R is discrete if it's image is countable. Def 2: X: Ω →R X: Ω → R is discrete if ∃E ⊆R ∃ E ⊆ R countable such that P(X−1(E)) = 1 P (X − 1 (E)) = 1. Now, 1 2 is clear to me. However I keep coming to ...
But if the random variable is discrete, then the CDF does not apply that? ( at least according to an exercise in a test ). Searched here for other topics of the same, could not find. Is there a different definition for continuous random variable, discrete random variable and not discrete\continuous random variable, for the CDF?
Yes, so, a discrete random variable is a measurable function from the sample space to the reals that can take an at most countably infinite number of values. By definition of random variable, all sets of the form X−1((−∞, t]) X − 1 ((− ∞, t]) are assigned a probability measure (for all t ∈R t ∈ R). For discrete random variables ...
2. Bounded, in this case, means what it means in pretty much every mathematical context - there's a maximum and minimum that the variable never exceeds. If you think about the normal random variable, Z, technically Z could be anything, it's just that as you get farther and farther away from zero, the probability diminishes exponentially.
After this huge eye opener Discrete Random Variables May Have Uncountable Images (cf 'Discrete Random Variables May Have Uncountable Images' - 'Almost surely countable'?) I checked my probability textbooks on their definitions of discrete random variable. From Rosenthal - A First Look at Rigorous Probability Theory