Linear Transformations - Data 140 Textbook

# HIDDEN
import warnings
warnings.filterwarnings('ignore')
from datascience import *
from prob140 import *
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
%matplotlib inline
import math
from scipy import stats

Linear transformations are both simple and ubiquitous: every time you change units of measurement, for example to standard units, you are performing a linear transformation.

16.1.1Linear Transformation: Exponential Density¶

Let $T$ have the exponential $(\lambda)$ distribution and let $T_1 = \lambda T$ . Then $T_1$ is a linear transformation of $T$ . Therefore

E(T_1) = \lambda E(T) = 1 ~~~ \text{and} ~~~ SD(T_1) = \lambda SD(T) = 1

The parameter $\lambda$ has disappeared in these results. Let’s see how that follows from the distribution of $T_1$ . The cdf of $T_1$ is

F_{T_1}(t) = P(T_1 \le t) = P(T \le t/\lambda) = 1 - e^{-\lambda (t/\lambda)} = 1 - e^{-t}

That’s the cdf of the exponential $(1)$ distribution, consistent with the expectation and SD we found above.

To summarize, if $T$ has the exponential $(\lambda)$ distribution then the distribution of $T_1 = \lambda T$ is exponential $(1)$ .

You can think of the exponential $(1)$ distribution as the fundamental member of the family of exponential distributions. All others in the family can be found by changing the scale of measurement, that is, by multiplying by a constant.

If $T_1$ has the exponential $(1)$ distribution, then $T = \frac{1}{\lambda}T_1$ has the exponential $(\lambda)$ distribution. The factor $1/\lambda$ is called the scale parameter.

Here are graphs of the densities of $T_1$ and $T = \frac{1}{2}T_1$ . By the paragraph above, $T$ has the exponential $(2)$ distribution.

t = np.arange(0, 5.01, 0.01)
t1 = stats.expon.pdf(t, scale = 1)
t2 = stats.expon.pdf(t, scale = 1/2)
plt.plot(t, t1, color='darkblue', lw=2, label='Exponential (1)')
plt.plot(t, t2, color='gold', lw=2, label='Exponential (2)')
plt.xlabel('$t$')
plt.ylabel('Density at $t$')
plt.legend();

The formulas for the two densities are

f_{T_1} (s) = e^{-s} ~~~~~~~~~~~~~~ f_T(t) = 2e^{-2t}

Let’s try to understand the relation between these two densities in a way that will help us generalize what we are seeing in this example.

The relation between the two random variables is $T = \frac{1}{2}T_1$ .

For any $t$ , the chance that $T$ is near $t$ is the same as the chance that $T_1$ is near $s = 2t$ . This explains the factor $e^{-2t}$ in the density of $T$ .
If we think of $T_1$ as a point on the horizontal axis, then to create $T$ you have to divide $T_1$ by 2. So the transformation consists of halving all distances on the horizontal axis. The total area under the density of $T$ must equal 1, so we have to compensate by doubling all distances on the vertical axis. This explains the factor 2 in the density of $T$ .

Answer

$\frac{\lambda}{c}$

from IPython.display import YouTubeVideo
YouTubeVideo('MA40E9QPCjU')

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16.1.2Linear Change of Variable Formula for Densities¶

We use the same idea to find the density of a linear transformation of a random variable.

Let $X$ be a random variable with density $f_X$ , and let $Y = aX + b$ for constants $a \ne 0$ and $b$ . Let $f_Y$ be the density of $Y$ . Then

f_Y(y) ~ = ~ f_X\big( \frac{y-b}{a}\big) \frac{1}{\lvert a \rvert}

Let’s take this formula in two pieces, as in the exponential example.

For $Y$ to be $y$ , $X$ has to be $(y-b)/a$ .
The linear function $y = ax+b$ involves multiplying distances along the horizontal axis by $\lvert a \rvert$ ; the sign of $a$ doesn’t affect distances. To get a density, we have to compensate by dividing all vertical distances by $\lvert a \rvert$ .

This is a good way to understand the formula, and will help you understand the corresponding formula for non-linear transformations.

For a formal proof, start with the case $a > 0$ .

F_Y(y) = P(aX+b \le y) = P\big(X \le \frac{y-b}{a}\big) = F_X\big(\frac{y-b}{a}\big)

By the chain rule of differentiation,

f_Y(y) = f_X\big(\frac{y-b}{a}\big) \cdot \frac{1}{a}

If $a < 0$ then division by $a$ causes the direction of the inequality to switch:

F_Y(y) = P(aX+b \le y) = P\big(X \ge \frac{y-b}{a}\big) = 1 - F_X\big(\frac{y-b}{a}\big)

Now the chain rule yields

f_Y(y) ~ = ~ -f_X\big(\frac{y-b}{a}\big) \cdot \frac{1}{a} ~ = ~ f_X\big(\frac{y-b}{a}\big) \cdot \frac{1}{\lvert a \rvert}

Answer

(a) For all $w$ , $f_W(w) = f\big(\frac{w-5}{3}\big)\cdot\frac{1}{3}$

(b) For all $w$ , $f_W(w) = f\big(\frac{w-5}{-3}\big)\cdot\frac{1}{3}$

16.1.3The Normal Densities¶

Let $Z$ have the standard normal density

\phi(z) ~ = ~ \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}z^2}, ~~~ -\infty < z < \infty

Let $X = \sigma Z + \mu$ for constants $\mu$ and $\sigma$ with $\sigma > 0$ . Then for any real number $x$ , the density of $X$ is

\begin{align*} f_X(x) ~ &= ~ \phi\big( \frac{x-\mu}{\sigma} \big) \frac{1}{\sigma} \\ \\ &= ~ \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2} \big( \frac{x-\mu}{\sigma} \big)^2} \end{align*}

Thus every normal random variable is a linear transformation of a standard normal variable.

Answer

$X = \sqrt{90}Z$ where $Z$ is $X$ in standard units and hence standard normal

16.1.4The Uniform Densities, Revisited¶

Let the distribution of $U$ be uniform on $(0, 1)$ and for constants $b > a$ let $V = (b-a)U + a$ . In an earlier section we saw that $V$ has the uniform distribution on $(a, b)$ . But let’s see what’s involved in confirming that result using our new formula.

First it is a good idea to be clear about the possible values of $V$ . Since the possible values of $U$ are in $(0, 1)$ , the possible values of $V$ are in $(a, b)$ .

At $v \in (a, b)$ , the density of $V$ is

f_V(v) ~ = ~ f_U\big( \frac{v - a}{b-a} \big) \frac{1}{b-a} ~ = ~ 1 \cdot \frac{1}{b-a} ~ = ~ \frac{1}{b-a}

That’s the uniform density on $(a, b)$ .

16.1 Linear Transformations

16.1.1Linear Transformation: Exponential Density¶

16.1.2Linear Change of Variable Formula for Densities¶

16.1.3The Normal Densities¶

16.1.4The Uniform Densities, Revisited¶