Familiar Scalar Location-Scale Families

Four Familiar Location-Scale Families

A location-scale family starts with a fixed standardized random variable \(Z\) and forms \(X=\mu+sZ\), where \(\mu\) is a location parameter and \(s>0\) is a scale parameter. Changing \(\mu\) shifts the distribution; changing \(s\) stretches or contracts it. The earlier material on location-scale transformations gives the general rules for the resulting distribution, density, support, moments, and quantiles. Here the emphasis is on recognizing those rules in four common scalar families and interpreting their different scale conventions.

The same numerical value of \(s\) need not mean the same standard deviation across families. In particular, a scale parameter is not automatically a standard deviation. The standardized base law determines how spread relates to \(s\), as well as the shape and support.

Standard Bases and Parameter Conventions

The following table fixes a base distribution and the resulting parameters for each family. For the normal family, the notation \(N(\mu,\sigma^2)\) uses variance as its second argument. For the other three families, \(s\) denotes their conventional scale parameter; the formulas in the table show how it relates to standard deviation.

The letters \(s\) and \(b\) are used here to emphasize that logistic and Laplace scales are family-specific conventions; neither is a standard deviation. The uniform scale \(s\) in this parameterization is the full width of its support. Another common uniform description uses a center \(\mu\) and half-width \(a\), so its support is \([\mu-a,\mu+a]\) and \(s=2a\).

The corresponding densities make the differences in shape explicit. The normal density is \(f(x)=(\sigma\sqrt{2\pi})^{-1}\exp(-(x-\mu)^2/(2\sigma^2))\). The logistic density is \(f(x)=s^{-1}e^{-(x-\mu)/s}/(1+e^{-(x-\mu)/s})^2\). The Laplace density is \(f(x)=(2b)^{-1}\exp(-|x-\mu|/b)\). The uniform density equals \(1/s\) on its support and zero outside it. These formulas use the density transformation stated in the definition.

Quantiles and Shape

All four families are symmetric about their location \(\mu\), and each has median \(\mu\). Their densities also attain their maximum at \(\mu\). Symmetry does not make their shapes interchangeable: the normal density has tails that decay at a Gaussian rate, while the logistic and Laplace tails decay exponentially. The Laplace density has a sharp peak at its center; the logistic density is smoother there. The uniform density is constant over a bounded interval and has no tails outside that interval.

For the normal distribution, quantiles are written using the standard normal quantile function: \(Q_X(p)=\mu+\sigma\Phi^{-1}(p)\), for \(0<p<1\). For the logistic family, the standard base CDF is \(F_Z(z)=1/(1+e^{-z})\). Its inverse is particularly simple:

To verify the logistic formula, solve \(p=1/(1+e^{-z})\). Rearranging gives \(1/p=1+e^{-z}\), hence \(e^{-z}=(1-p)/p\), so \(z=\log(p/(1-p))\). Since \(X=\mu+sZ\) with \(s>0\), its \(p\)-quantile is \(\mu+s\) times the base quantile in the affine sense: \(Q_X(p)=\mu+sQ_Z(p)\). Substitution gives the stated expression. For the uniform base, \(F_Z(z)=z+1/2\) on \([-1/2,1/2]\), so \(Q_Z(p)=p-1/2\), yielding the displayed uniform quantile. \(\square\)

Moments from the Base Distributions

The moment formulas in the table can be checked directly for the uniform and Laplace bases. The derivations also illustrate why the standardized base determines how scale translates into spread.

For the Laplace law, symmetry about \(\mu\) gives \(E[X]=\mu\). With \(Z=(X-\mu)/b\), the standardized density is \(\tfrac12e^{-|z|}\). Its mean is zero, and \(E[Z^2]=\int_0^\infty z^2e^{-z}\,dz=2\). The last integral follows by integration by parts twice: \(\int_0^\infty z^2e^{-z}\,dz=2\int_0^\infty ze^{-z}\,dz=2\int_0^\infty e^{-z}\,dz=2\); the boundary terms vanish because \(z^ke^{-z}\to0\) as \(z\to\infty\) for \(k=1,2\). Therefore \(\operatorname{Var}(X)=b^2\operatorname{Var}(Z)=2b^2\), and the standard deviation is \(b\sqrt{2}\). The logistic variance \(s^2\pi^2/3\) is the standard logistic moment formula; its scale-to-standard-deviation conversion is shown in the table.

How to Choose and Report a Parameterization

These families are useful as compact descriptions of different distributional shapes. A normal model is a familiar symmetric, unbounded benchmark. A logistic model has heavier tails than the normal and a convenient quantile formula. A Laplace model has a sharper central peak and heavier tails than the normal. A uniform model describes constant density over a bounded interval. Selecting one is a modeling decision: the location-scale representation does not itself establish that a particular family fits data.

When reporting a model, give both the family and the parameter convention. For example, “uniform with center 10 and full width 12” identifies support \([4,16]\); “uniform on \([4,16]\)” states the support directly. For logistic and Laplace models, specify whether the reported spread is the family scale or the standard deviation. For a normal model, state whether the second argument is variance, as in \(N(\mu,\sigma^2)\).

Common Mistakes

• Calling every scale a standard deviation. For example, Laplace scale \(b=3\) gives standard deviation \(3\sqrt{2}\), not 3. Use the family-specific conversion.
• Confusing uniform width and half-width. In this tutorial \(s\) is full support width, so the endpoints are \(\mu\pm s/2\). A half-width parameter \(a\) instead gives endpoints \(\mu\pm a\).
• Reading the normal parameter incorrectly. In \(N(\mu,\sigma^2)\), the second argument is variance, while \(\sigma\) is the standard deviation.
• Assuming equal location and scale imply equal shapes. Normal, logistic, and Laplace distributions all have support \(\mathbb{R}\), but their peak and tail behavior differ; the uniform law even has bounded support.