PDF/CDF Visualizer
Choose a named continuous random variable and adjust its parameters. The plots of its probability density function and cumulative distribution function update as you go, and the buttons below the plots open tools for computing probabilities and the expected value.
Random Variable
Parameters
Probability density function
Cumulative distribution function
The probability density function (PDF), \(f_X(x)\), describes how probability is spread along the number line: the probability that a continuous random variable \(X\) lands in an interval is the area under \(f_X\) over that interval. The cumulative distribution function (CDF), \(F_X(x) = P(X \le x) = \int_{-\infty}^{x} f_X(t)\,dt\), gives the probability that \(X\) is at most \(x\); it is the running area under the PDF. The two plots above update together, and any probability you compute is marked in pink in both.
Compute probability
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Pink marks the event: on the left, the area under the PDF over the event; on the right, the part of the CDF over the event, which rises from the lower dashed level to the upper one.
Remember: For a continuous random variable, \(P(X = x) = 0\) for every single value \(x\). Probability is area under the density curve, and a single point is an interval of width zero, so it has no area. As a result, endpoints never matter: \(P(a \le X \le b) = P(a < X < b)\). The density \(f_X(x)\) is not itself a probability, so it can be larger than 1.
Expected value
Blue marks E[X]: on the left, the balance point of the PDF, with a fulcrum under the axis; on the right, a dashed line at the same x.