Why is it P(X=a) = 0 for all a belonging to ℝ for continuus rand.vars?

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SUMMARY

The probability P(X=a) equals 0 for all a in ℝ due to the properties of continuous random variables. This is established through the probability density function (PDF), where P(X ∈ [a, b]) is calculated as the integral of f(x) from a to b. Specifically, P(X=a) is represented as P(X ∈ [a, a]), which simplifies to the integral from a to a, resulting in 0. This confirms that individual points in a continuous distribution have no probability mass.

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cdux
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I probably miss something basic, unless it's an abstract definition.
 
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[tex]P(X\in [a, b])= \int_a^b f(x)dx[/tex]
where f(x) is the "probability density function".

P(X= a) means [tex]P(X \in [a,a])= \int_a^a f(x)dx= 0[/tex]
 
Thank you. I guess it's like my initial assumption: "it represents an atomic elelement of the integral and that tends to 0".
 

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