Introduction

Linear Transformation

Consider the linear transformation,

\[ Y = aX + b \]

then the PDF of \(Y\) follows

\[ f_Y(y) = \frac{1}{\abs{a}}f_X\left(\frac{y-b}{a}\right) \]

Proof (for \(a>0\)) : For any value \(y\) in the range of \(Y\), the CDF of \(Y\) is then

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

By chain rule,

$$
\begin{align*}
f_Y(y) &= \frac{dF_Y}{dy} \\
&= \frac{dF_Y}{dx}\frac{dx}{dy}\\
&= f_X\left(\frac{y-b}{a}\right)\frac{1}{a}\\
\end{align*}
$$

Proof (for \(a<0\)) : Using the same prove as before, you can see that only the inequality sign flips which we can take the complement where the derivative ignores the complement.

Examples

• Normal Density

  For \(Z\) is the standard normal random variable, \(X\) is the \(\text{Normal}(\mu, \sigma)\) which is the linear transformation of \(Z\),

  \[ X = \sigma Z + \mu \]

• Uniform Density

  For \(U_1\) is \(\text{Uniform}(0,1)\) and \(U\) is \(\text{Uniform}(a,b)\), then \(U\) is the linear transformation of \(U_1\),

  \[ U = (b-a)U_1 + a \]

Monotonic Transformation

Let’s begin by witha motivating example. Given uniform function \(U \sim \text{Uniform}(0,1)\). and a monotonically increasing function \(F\), we define \(U\) to be the monotonically increasing trasnformation of \(X\),

\[ X = F^{-1}(U) \]

Then CDF of \(X\) is,

\[ F_X(x) = F(x) \]

In general let \(g\) be a monotonically increasing function such that for any random variable \(X\),

\[ Y = g(X) \]

Then the CDF of \(Y\) is,

\[ \boxed{F_Y(y) = F_X(g^{-1}(y))} \]

Thus the PDF of \(Y\) is,

\[ \boxed{f_Y(y) = \frac{f_X(x)}{g'(x)}; \qquad x = g^{-1}(y)} \]

\[\begin{split} \begin{align*} f_Y(y) &= \frac{dF_Y}{dy}\\ &= \frac{dF_Y}{dx}\frac{dx}{dy}\\ &= \frac{f_X(x)}{g'(x)} \tag{Derivative of Inverse Functions} \end{align*} \end{split}\]

Bitonic Transformation

The name Bitonic isn’t convention name so let’s define it. A bitonically increasing function a function \(g(x)\) that the absolute value of \(g(x)\) increase as \(x\) increases for the function \(g(x_0 + x)\) where \(g(x_0)\) is some global minima.

Two-to-One Transformation

A Two-to-One function is a function \(g(x)\) that for every value \(y\), there exists two \(x\) that satisfies \(g(x) = y\). An example of a two-to-one function is \(g(x)=x^2\) since to the right of \(x_0=0\), the function is monotinically increasing and to the left the function is monotonically decreasing. Thus, the absolute value of \(g(x)\) is increasing on both sides.

Using the square as the motivating example let,

\[ Y = X^2 \]

That implies that the PDF of \(Y\) must consider when \(X=\sqrt{y}\) and \(X=-\sqrt{y}\) where we assign those values to be \(x_1\) and \(x_2\) respectively. Since both sides are monotonically increasing functions, we can apply monotonic transformation for the PDF to be,

\[\begin{split} \begin{align*} f_Y(y) &= \frac{f_X(x_1)}{g'(x_1)} + \frac{f_X(x_2)}{g'(x_2)}\\ &= \frac{f_X(x_1)}{2x_1} + \frac{f_X(x_2)}{\abs{2x_2}} \end{align*} \end{split}\]