Linear Transformation of Continuous Random Variable is Continuous Random Variable

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:

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:

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


 * $F_{a X + b}$ is continuous

in the case $a < 0$.