Why is P(X=c)=0 for any number c in the domain of a probability density function?

Name: Problem 28, Chapter 11: Calculus with Applications
Uploaded: 2019-04-23T13:35:35-08:00
Duration: 3 min 18 s
Channel: Matt Just
Description: Why is P(X=c)=0 for any number c in the domain of a probability density function?

step 1 Okay, so the question is why is the probability that x is equal to a fixed real number c equal t

Why is P(X=c)=0 for any number c in the domain of a...

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Key Concepts

Integration Over Intervals

For continuous random variables, probabilities are calculated by integrating the pdf over an interval. This process sums up the infinitesimal contributions of all points in the interval, yielding a finite probability, whereas a single point, having no interval width, does not contribute any probability.

Measure Zero

The concept of measure zero comes from measure theory, where any single point in a continuous space is considered to have no length, area, or volume. Consequently, when calculating probabilities via integration over the entire space, each individual point contributes zero to the total probability.

Continuous Random Variable

A continuous random variable is one that can take on an infinite number of possible values within a given range. Since there are infinitely many such values, the probability of the variable taking on any specific, exact value is infinitesimally small, effectively zero.

Probability Density Function

A probability density function (pdf) describes the relative likelihood of a continuous random variable taking on a particular value. However, the pdf itself does not provide probabilities directly; instead, probabilities are obtained by integrating the pdf over a range of values. This is why the value of the pdf at a single point does not represent a probability.

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