Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2

Summary by Krater

This video explains the concept of probability density and probability density functions in continuous probability distributions, resolving the paradox of assigning individual probabilities to infinite values in a continuous range.

From the video

Answers: What is probability density and how do you work with probabilities of continuous values?

• probability density
• probability density function
• continuous random variables
• measure theory
• Riemann integration
• Lebesgue integration

What it concludes

• When dealing with continuous probability distributions, individual values have a probability of zero, and probabilities are instead assigned to ranges using a probability density function.
• The total area under a probability density function curve must always equal one.
• The probability of a random variable falling between two values in a continuous distribution equals the area under the probability density curve between those values.
• Probability density represents probability per unit in the x-direction.
• Measure theory provides a rigorous mathematical foundation that unites discrete and continuous probability settings.