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

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In summary, for a continuous random variable, the probability of any specific value occurring is infinitesimally small, resulting in a cumulative probability of 0 for all values on the real number line. This is due to the infinite number of possible outcomes for continuous random variables. The probability of a continuous random variable is calculated by finding the area under the probability density function (PDF) curve, which has a total area of 1. Understanding that the probability of a continuous random variable taking on any specific value is 0 is important for interpreting and analyzing data. While there may be exceptions, such as in discrete distributions, the probability of a continuous random variable taking on any single point or value is generally considered to be 0.
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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]
 
  • #3
Thank you. I guess it's like my initial assumption: "it represents an atomic elelement of the integral and that tends to 0".
 

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

Why is the probability of a continuous random variable equal to 0 for all values on the real number line?

For a continuous random variable, the probability of any single point or value occurring is infinitesimally small. This means that the probability of the random variable taking on a specific value is essentially 0. In other words, the probability of any specific point on the real number line is equal to 0, resulting in a cumulative probability of 0 for all values on the real number line.

Can the probability of a continuous random variable ever be greater than 0?

No, the probability of a continuous random variable cannot be greater than 0 for any single point or value. This is due to the nature of continuous random variables, which have an infinite number of possible outcomes. As a result, the probability of any single outcome is infinitesimally small, approaching 0.

How is the probability of a continuous random variable calculated?

The probability of a continuous random variable is calculated by finding the area under the probability density function (PDF) curve. This is also known as the integral of the PDF over a given interval. The total area under the curve is equal to 1, representing the total probability of all possible outcomes.

Why is it important to understand that P(X=a) = 0 for all a belonging to ℝ for continuous random variables?

Understanding that the probability of a continuous random variable taking on any specific value is 0 is important for interpreting and analyzing data. It allows us to calculate the probability of a range of values, rather than a single point. Additionally, it helps us understand the concept of continuous probability distributions and their properties.

Are there any exceptions to P(X=a) = 0 for continuous random variables?

There are some cases where the probability of a continuous random variable taking on a specific value may not be exactly 0. For example, if the continuous random variable follows a discrete distribution, the probability of specific values may be non-zero. However, in general, the probability of a continuous random variable taking on any single point or value is considered to be 0.

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