PDF/CDF Visualizer
Pick a random variable and move its sliders; the PDF and CDF update together. The buttons below the plots open more tools, each with its own sliders.
Random Variable
Parameters
Probability density function
Cumulative distribution function
PDF: \(f_X(x)\); the area under it over an interval is a probability. CDF: \(F_X(x) = P(X \le x)\), the running area under the PDF.
Compute probability
=
PDF with the event highlighted
Remember: A single point has zero probability, so endpoints don't matter: \(P(a \le X \le b) = P(a < X < b)\). The density can exceed 1; only areas are probabilities.
Expected value
Variance
Var(X) = σ =
Skewness
Skewness γ1 =
Cubing keeps the sign of each deviation, and dividing by \(\sigma^{3}\) removes the units.
Kurtosis
Excess kurtosis γ2 = Kurt(X) =
The fourth power makes values far from the mean count heavily, so kurtosis is mostly about the tails.
Why “excess”: every normal distribution has kurtosis 3, so \(\gamma_2 = \operatorname{Kurt}(X) - 3\) puts the normal at zero.
Moments
MX(t) =
Evaluate the MGF at