Inverse CDF Method Visualized

F^-(u) = inf{x : F(x) ≥ u} — Works for ALL distributions!

Distribution Type:

u value: 0.50

Animate u from 0 to 1

Transform: u = 0.50 → F^-(u) = 0.00

💡 Key Insight

For continuous distributions, the CDF is strictly increasing, so there's a unique x where F(x) = u. The steeper the CDF, the higher the density at that point.

PDF / PMF: f(x)

Probability density or mass

CDF: F(x)

Cumulative probability

Inverse CDF: F^-(u)

Quantile function

📈 PDF/PMF Panel

Shows where probability mass lives
Shaded area = P(X ≤ F^-(u)) = u
Discrete: bars show P(X = k)

📊 CDF Panel

Draw horizontal line at height u
Find where it hits the CDF
Discrete: jumps create "flat regions"

📋 Inverse CDF Panel

Direct view of u → x mapping
Discrete: plateaus (many u → same x)
This is what we compute!