Probability density function

Definition

A probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. The area under the PDF curve over an interval represents the probability of the variable falling within that interval.

5 Must Know Facts For Your Next Test

1. The integral of a PDF over its entire range is equal to 1.
2. A PDF must be non-negative for all possible values of the random variable.
3. Improper integrals are often used when calculating probabilities involving PDFs with infinite bounds.
4. The mean (expected value) of a continuous random variable can be found by integrating the product of the variable and its PDF over all possible values.
5. A cumulative distribution function (CDF) is obtained by integrating the PDF from negative infinity to a specific value.

Review Questions

• What is the integral of a probability density function over its entire range?
• How do you find the mean of a continuous random variable using its PDF?
• Explain how improper integrals relate to probability density functions.

Related terms

Cumulative Distribution Function (CDF): A function representing the probability that a continuous random variable takes on a value less than or equal to a specific point, obtained by integrating the PDF up to that point.

Expected Value: The mean or average value of a random variable, calculated as an integral involving its PDF.

Improper Integral:

An integral with at least one infinite limit or an integrand that approaches infinity at some points within the integration range.