Show That Exponential Is a Member of Scale Family Distributions

5.one: Location-Scale Families

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• Professor Emeritus (Mathematics) at University of Alabama in Huntsville
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\(\newcommand{\P}{\mathbb{P}}\) \(\newcommand{\R}{\mathbb{R}}\) \(\newcommand{\East}{\mathbb{Due east}}\) \(\newcommand{\N}{\mathbb{N}}\) \(\newcommand{\var}{\text{var}}\) \(\newcommand{\sd}{\text{sd}}\) \(\newcommand{\skw}{\text{skew}}\) \(\newcommand{\kur}{\text{kurt}}\)

General Theory

As usual, our starting point is a random experiment modeled past a probability infinite \( (\Omega, \mathscr F, \P) \), and then that \( \Omega \) is the set of outcomes, \( \mathscr F \) the collection of events, and \( \P \) the probability mensurate on the sample infinite \( (\Omega, \mathscr F) \). In this section, we assume that nosotros fixed random variable \( Z \) defined on the probability infinite, taking values in \( \R \).

Definition

For \(a \in \R\) and \(b \in (0, \infty) \), let \(X = a + b \, Z\). The two-parameter family of distributions associated with \(X\) is called the location-scale family unit associated with the given distribution of \(Z\). Specifically, \(a\) is the location parameter and \(b\) the scale parameter.

Thus a linear transformation, with positive gradient, of the underlying random variable \(Z\) creates a location-scale family for the underlying distribution. In the special instance that \(b = 1\), the 1-parameter family is called the location family associated with the given distribution, and in the special instance that \(a = 0\), the one-parameter family unit is called the scale family associated with the given distribution. Scale transformations, as the name suggests, occur naturally when physical units are changed. For example, if a random variable represents the length of an object, and then a change of units from meters to inches corresponds to a scale transformation. Location transformations often occur when the zero reference point is changed, in measuring distance or time, for example. Location-scale transformations can also occur with a modify of physical units. For example, if a random variable represents the temperature of an object, so a change of units from Fahrenheit to Celsius corresponds to a location-scale transformation.

Distribution Functions

Our goal is to relate various functions that make up one's mind the distribution of \( X = a + b Z \) to the corresponding functions for \( Z \). Showtime we consider the (cumulative) distribution function.

If \(Z\) has distribution function \(G\) then \(X\) has distribution function \(F\) given by \[ F(x) = G \left( \frac{x - a}{b} \right), \quad x \in \R\]

Proof

For \( ten \in \R \) \[ F(x) = \P(X \le ten) = \P(a + b Z \le x) = \P\left(Z \le \frac{10 - a}{b}\right) = M\left(\frac{x - a}{b}\right) \]

Next we consider the probability density function. The results are a bit different for discrete distributions and continuous distribution, not surprising since the density office has dissimilar meanings in these two cases.

If \( Z \) has a discrete distribution with probability density function \( grand \) and then \( X \) likewise has a discrete distribution, with probability density function \( f \) given by \[ f(x) = thousand\left(\frac{x - a}{b}\correct), \quad x \in \R \]

Proof

\( Z \) takes values in a countable subset \( S \subset \R \) and hence \( X \) takes values in \( T = \{a + b z: z \in S\} \), which is also countable. Moreover \[ f(10) = \P(X = x) = \P\left(Z = \frac{x - a}{b}\right) = one thousand\left(\frac{x - a}{b}\correct), \quad x \in \R \]

If \(Z\) has a continuous distribution with probability density function \(g\), and so \(X\) also has a continuous distribution, with probability density function \(f\) given past

\[ f(x) = \frac{i}{b} \, g \left( \frac{x - a}{b} \right), \quad x \in \R\]

1. For the location family associated with \(g\), the graph of \(f\) is obtained by shifting the graph of \(g\), \(a\) units to the right if \(a \gt 0\) and \(-a\) units to the left if \(a \lt 0\).
2. For the scale family associated with \(m\), if \(b \gt 1\), the graph of \(f\) is obtained from the graph of \(g\) by stretching horizontally and compressing vertically, by a factor of \(b\). If \(0 \lt b \lt one\), the graph of \(f\) is obtained from the graph of \(yard\) past compressing horizontally and stretching vertically, by a factor of \(b\).

Proof

First note that \( \P(X = x) = \P\left(Z = \frac{x - a}{b}\correct) = 0 \), so \( X \) has a continuous distribution. Typically, \( Z \) takes values in an interval of \( \R \) and thus so does \( X \). The formula for the density office follows past taking derivatives of the distribution office above, since \( f = F^\prime \) and \( g = G^\prime \).

If \(Z\) has a style at \(z\), then \(Ten\) has a mode at \(x = a + b z\).

Proof

This follows from density function in the detached instance or the density function in the continuous case. If \( g \) has a maximum at \( z \) and then \( f \) has a maximum at \( x = a + b z \)

Adjacent we relate the quantile functions of \(Z\) and \(X\).

If \(One thousand\) and \(F\) are the distribution functions of \(Z\) and \(10\), respectively, then

1. \(F^{-1}(p) = a + b \, Chiliad^{-1}(p)\) for \(p \in (0, i)\)
2. If \(z\) is a quantile of lodge \(p\) for \(Z\) so \(x = a + b \, z\) is a quantile of order \(p\) for \(X\).

Proof

These results follow from the distribution function above.

Suppose at present that \( Z \) has a continuous distribution on \([0, \infty)\), and that nosotros think of \(Z\) as the failure time of a device (or the fourth dimension of death of an organism). Permit \(X = b Z\) where \( b \in [0, \infty)\), and then that the distribution of \(X\) is the calibration family associated with the distribution of \(Z\). Then \(10\) also has a continuous distribution on \([0, \infty)\) and can also be thought of as the failure time of a device (perhaps in different units).

Allow \(One thousand^c\) and \(F^c\) denote the reliability functions of \(Z\) and \(10\) respectively, and permit \(r\) and \(R\) denote the failure charge per unit functions of \(Z\) and \(X\), respectively. And so

1. \(F^c(x) = G^c(x / b)\) for \(x \in [0, \infty)\)
2. \(R(x) = \frac{1}{b} r\left(\frac{10}{b}\right)\) for \(x \in [0, \infty)\)

Proof

Recall that \( Thousand^c = one - G \), \( F^c = i - F \), \( r = g / \bar{G} \), and \( R = f / \bar{F} \). Thus the results follow from the distribution role and the density office above.

Moments

The post-obit theorem relates the mean, variance, and standard divergence of \(Z\) and \(X\).

Every bit earlier, suppose that \(X = a + b \, Z\). And so

1. \(\E(X) = a + b \, \E(Z)\)
2. \(\var(X) = b^2 \, \var(Z)\)
3. \(\sd(Ten) = b \, \sd(Z)\)

Proof

These upshot follow immediately from basic backdrop of expected value and variance.

Recall that the standard score of a random variable is obtained by subtracting the mean and dividing past the standard difference. The standard score is dimensionless (that is, has no physical units) and measures the distance from the hateful to the random variable in standard deviations. Since location-scale familes essentially represent to a alter of units, it'due south non surprising that the standard score is unchanged by a location-scale transformation.

The standard scores of \(Ten\) and \(Z\) are the same:

\[ \frac{Ten - \E(X)}{\sd(X)} = \frac{Z - \E(Z)}{\sd(Z)} \]

Proof

From the mean and variance in a higher place:

\[ \frac{X - \E(10)}{\sd(X)} = \frac{a + b Z - [a + b \E(Z)]}{b \sd(Z)} = \frac{Z - \E(Z)}{\sd(Z)} \]

Recall that the skewness and kurtosis of a random variable are the third and fourth moments, respectively, of the standard score. Thus it follows from the previous result that skewness and kurtosis are unchanged by location-scale transformations: \(\skw(10) = \skw(Z)\), \(\kur(10) = \kur(Z)\).

Nosotros can represent the moments of \( 10 \) (about 0) to those of \( Z \) past means of the binomial theorem: \[ \East\left(10^n\correct) = \sum_{yard=0}^n \binom{northward}{k} b^1000 a^{n - one thousand} \E\left(Z^chiliad\correct), \quad north \in \N \] Of course, the moments of \( X \) about the location parameter \( a \) have a uncomplicated representation in terms of the moments of \( Z \) about 0: \[ \East\left[(X - a)^north\right] = b^n \E\left(Z^northward\correct), \quad n \in \North \] The following practise relates the moment generating functions of \(Z\) and \(Ten\).

If \(Z\) has moment generating part \(m\) then \(10\) has moment generating part \(M\) given by

\[ 1000(t) = e^{a t} m(b t) \]

Proof

\[ Thou(t) = \Eastward\left(e^{tX}\right) = \E\left[e^{t(a + bZ)}\right] = e^{ta} \E\left(east^{t b Z}\right) = e^{a t} m(b t) \]

Type

As we noted earlier, two probability distributions that are related past a location-scale transformation tin can be thought of equally governing the aforementioned underlying random quantity, but in different concrete units. This human relationship is important enough to deserve a name.

Suppose that \( P \) and \( Q \) are probability distributions on \( \R \) with distribution functions \(F\) and \(G\), respectively. Then \( P \) and \( Q \) are of the same type if in that location exist constants \(a \in \R\) and \(b \in (0, \infty)\) such that \[ F(x) = One thousand \left( \frac{x - a}{b} \right), \quad x \in \R \]

Being of the aforementioned type is an equivalence relation on the drove of probability distributions on \(\R\). That is, if \(P\), \(Q\), and \(R\) are probability distribution on \( \R \) and so

1. \(P\) is the same type as \(P\) (the reflexive holding).
2. If \(P\) is the same type as \(Q\) then \(Q\) is the same type equally \(P\) (the symmetric property).
3. If \(P\) is the same type as \(Q\), and \(Q\) is the same blazon as \(R\), then \(P\) is the aforementioned type as \(R\) (the transitive property).

Proof

Let \( F \), \( G \), and \( H \) denote the distribution functions of \( P \), \( Q \), and \( R \) respectively.

1. This is picayune, of course, since nosotros tin can take \( a = 0 \) and \( b = one \).
2. Suppose there exists \( a \in \R \) and \( b \in (0, \infty) \) such that \( F(ten) = G\left(\frac{x - a}{b}\right) \) for \( 10 \in \R \). Then \( G(x) = F(a + b ten) = F\left(\frac{x - (-a/b)}{1/b}\right) \) for \( x \in \R \).
3. Suppose there exists \( a, \, c \in \R \) and \( b, \, d \in (0, \infty) \) such that \( F(x) = Thou\left(\frac{10 - a}{b}\right) \) and \( G(x) = H\left(\frac{x - c}{d}\right) \) for \( x \in \R \). Then \( F(x) = H\left(\frac{10 - (a + bc)}{bd}\right)\) for \( x \in \R \).

Then, the collection of probability distributions on \( \R \) is partitioned into mutually exclusive equivalence classes, where the distributions in each class are all of the aforementioned type.

Examples and Applications

Special Distributions

Many of the special parametric families of distributions studied in this chapter and elsewhere in this text are location and/or calibration families.

The gamma distribution is a scale family for each value of the shape parameter.

The Pareto distribution is a scale family unit for each value of the shape parameter.

The triangle distribution is a location-scale family for each value of the shape parameter.

The U-power distribution is a location-scale family for each value of the shape parameter.

The Weibull distribution is a scale family for each value of the shape parameter.

The Wald distribution is a calibration family unit, although in the usual formulation, neither of the parameters is a scale parameter.

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Source: https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.01%3A_Location-Scale_Families

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