PMF/CDF Visualizer
Pick a random variable and move its sliders; the PMF and CDF update together. The buttons below the plots open more tools, each with its own sliders.
Random Variable
Parameters
Probability mass function
Cumulative distribution function
PMF: \(P_X(x) = P(X = x)\). CDF: \(F_X(x) = P(X \le x)\), the running sum of the PMF.
Compute probability
=
PMF with the event highlighted
Remember: Only whole numbers count, so \(P(X \le 2.7) = P(X \le 2)\).
Expected value
E[X] =
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