Probability Mass and Density Functions: A Formal Description A probability mass function (PMF) assigns a non-zero probability measure to each element in...
A probability mass function (PMF) assigns a non-zero probability measure to each element in a discrete probability space. In other words, it tells us the probability of a specific event occurring. The sum of probabilities over all possible outcomes in a probability space must always equal 1.
The probability density function (PDF) assigns a non-zero probability density to each point in a continuous probability space. This means that the PDF tells us the probability density of finding a random variable with that specific value. The total probability under the PDF must equal 1 over the entire space.
Here's a simple analogy to help differentiate between PMFs and PDFs:
PMF: Imagine throwing a single dart at a dartboard. The PMF would tell you the probability of the dart landing at different points on the board, based on the size and arrangement of the dartboard's markings.
PDF: Imagine measuring the height of women in a population. The PDF would tell you the probability density of finding a woman with a specific height, based on the population's height distribution.
Key Differences:
PMF: PMFs are used when we have a finite number of possible outcomes. For continuous spaces, we use PDFs.
Range: A PMF assigns a probability measure, while a PDF assigns a probability density.
Integration: Calculating the probability of an event in a probability space involves summing the PMF values over all possible outcomes. Integration for PDFs involves finding the total probability under the PDF.
Examples: