Density and cumulative shown side-by-side. Area under the PDF equals height of the CDF.
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CDF vs PDF
This diagram places the probability density function (PDF) and the cumulative distribution function (CDF) for the same distribution one above the other, sharing an x-axis. The PDF (top panel) shows how probability is concentrated across values. The CDF (bottom panel) shows, for any point x, what fraction of outcomes fall at or below that value. Moving a vertical cursor to x = 1.0 on a standard normal, for instance, shades the left tail of the PDF and simultaneously marks the CDF height at roughly 0.84, confirming that 84% of values lie below 1.0.\n\nThe paired display makes the integral relationship concrete: the shaded area under the PDF equals the height of the CDF at that same x. This is the visual proof that the CDF is the running integral of the PDF.
The PDF answers "how dense is the probability near this value?" and can exceed 1.0 for continuous distributions as long as the total area integrates to 1. The CDF answers "what fraction of outcomes fall below this value?" and always rises monotonically from 0 to 1.\n\nPractically, the CDF is more useful for probability questions: to find P(a < X < b) from the CDF, subtract F(a) from F(b). Percentiles read directly off the CDF: the 75th percentile is the x where the CDF reaches 0.75. The PDF is more intuitive for seeing where probability is concentrated and for comparing distribution shapes, but requires an integral to convert any density value into an actual probability.
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This cdf vs pdf is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
Density and cumulative representations
The probability density function (PDF) shows the likelihood of observing a value at each point on the x-axis; the cumulative distribution function (CDF) shows the total probability of observing a value less than or equal to each x. For a continuous distribution, the area under the PDF equals the height of the CDF.
A side-by-side view makes this relationship explicit: the PDF's bell shape transforms into the CDF's S-shape. Where the PDF peaks, the CDF rises most steeply; where the PDF flattens in the tails, the CDF levels off toward 0 or 1.
Practical applications of each form
The PDF is intuitive for understanding the most likely values and the distribution shape. The CDF is more direct for answering questions like 'what is the probability of observing a value below x?' or 'at what value does 90 percent of the distribution fall below?'
Statistical software typically computes both: the PDF for density estimates and histograms, the CDF for quantile calculations and tail probability assessments. Understanding both forms allows you to convert between questions about likelihood and questions about cumulative probability.
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