Linear Transformation of Continuous Random Variable is Continuous Random Variable

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Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $a$ be a non-zero real number.

Let $b$ be a real number.

Let $X$ be a continuous real variable.

Let $F_X$ be the cumulative distribution function of $X$.


Then $a X + b$ is a continuous real variable.


Further, if $a > 0$, the cumulative distribution function of $a X + b$, $F_{a X + b}$. is given by:

$\ds \map {F_{a X + b} } x = \map {F_X} {\frac {x - b} a}$

for each $x \in \R$.

If $a < 0$, $F_{a X + b}$ is given by:

$\ds \map {F_{a X + b} } x = 1 - \map {F_X} {\frac {x - b} a}$

for each $x \in \R$.


Proof

From Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable, $a X + b$ is a real-valued random variable.

Since $X$ is a continuous real variable, we have that:

$F_X$ is continuous.

We use this fact to show that $F_{a X + b}$ is continuous, showing that $a X + b$ is a continuous real variable.

We split up into cases.


Suppose that $a > 0$.

Then, for each $x \in \R$, we have:

\(\ds \map {F_{a X + b} } x\) \(=\) \(\ds \map \Pr {a X + b \le x}\)
\(\ds \) \(=\) \(\ds \map \Pr {a X \le x - b}\)
\(\ds \) \(=\) \(\ds \map \Pr {X \le \frac {x - b} a}\)
\(\ds \) \(=\) \(\ds \map {F_X} {\frac {x - b} a}\) Definition of Cumulative Distribution Function

From Composite of Continuous Mappings is Continuous and Linear Function is Continuous, we therefore have:

$F_{a X + b}$ is continuous

in the case $a > 0$.


Now suppose that $a < 0$.

Then, for each $x \in \R$, we have:

\(\ds \map {F_{a X + b} } x\) \(=\) \(\ds \map \Pr {a X + b \le x}\)
\(\ds \) \(=\) \(\ds \map \Pr {a X \le x - b}\)
\(\ds \) \(=\) \(\ds \map \Pr {X \ge \frac {x - b} a}\)
\(\ds \) \(=\) \(\ds \map \Pr {\Omega \setminus \set {X < \frac {x - b} a} }\)
\(\ds \) \(=\) \(\ds 1 - \map \Pr {X < \frac {x - b} a}\) Probability of Event not Occurring
\(\ds \) \(=\) \(\ds 1 - \map \Pr {X \le \frac {x - b} a}\) Probability of Continuous Random Variable Lying in Singleton Set is Zero
\(\ds \) \(=\) \(\ds 1 - \map {F_X} {\frac {x - b} a}\)

From Composite of Continuous Mappings is Continuous and Linear Function is Continuous, we therefore have:

$F_{a X + b}$ is continuous

in the case $a < 0$.

$\blacksquare$

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