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Normal loss density under standardization

For σL>0, fL(l)=φ((l−μL)/σL)/σL.

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Define the coordinate transformation

Standardized loss is z=(l−μL)/σL. An interval dl in loss units corresponds to dz=dl/σL. Probability area must be preserved when the horizontal coordinate changes.

Apply the inverse scale

fL(ℓ)=1σLϕ ⁣(ℓ−μLσL),σL>0f_L(\ell)=\frac{1}{\sigma_L}\phi\!\left(\frac{\ell-\mu_L}{\sigma_L}\right),\qquad\sigma_L>0

The density is probability per loss unit, rather than probability at a point.

Compare original density heights

At μL=0 and σL=10, loss density at zero is about 0.039894 per loss unit, whereas standard normal density at z=0 is about 0.398942 per z unit. The factor ten difference reflects a horizontal scale change, not lost probability.

Read the plotted basis

The calculator uses z coordinates and standard density, so its curve stays the same when loss-unit scale changes. Tail output amounts still depend on the entered location and dispersion. No density plot is defined by this expression at zero dispersion.

Further references