1 · The normal density, with adjustable μ and σ

Drag the sliders: the curve shifts with μ and spreads with σ. The shaded band is the probability of falling between the two cut points a and b — it is the area under the density, computed with the error function.

0.0 1.00 -1.0 1.0
P(a < X ≤ b) = 0.6827  ·  F(b) − F(a) = 0.6827

2 · Change of variables: Y = X², X ~ N(0,1)

Left: the standard normal density fX, with the two preimages ±√y of a point y marked. Right: the pushed-forward density fY(y) = (f(√y) + f(−√y)) / (2√y) — the mass of two symmetric strips, divided by the stretched width 2√y. Move the slider to see how a strip of X-mass maps to a strip of Y-mass.

1.00
fY(y) = 0.1995  ·  mass in strip of X ≈ 0.3413  ·  strip width 2√y = 2.00