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.
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The integral of a PDF over its entire range is equal to 1.
A PDF must be non-negative for all possible values of the random variable.
Improper integrals are often used when calculating probabilities involving PDFs with infinite bounds.
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.
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.