What all applications of Linear transformation of Random Variable

Question

Imagine we have a random variable X, we can apply a linear transformation to it by Y = c + aX, this will just shift and scale the distribution. In practical applications what all scenarios this transformation are used.

Answer 1

The most common scenario is normalizing a random variable. That is, given $X$, define $Y$ by $$ Y=\frac{X-\mathbb{E}X}{\sqrt{\operatorname{Var}(X)}}. $$ Then, $\mathbb{E}Y=0$ and $\operatorname{Var}Y=1$.

Answer 2

Various scenarios are conceivable. One scenario could be the following: Consider you have a fair 6 sides die and you want to play with a die that has the outcomes $5,7,9,11,13,15$. Then you can make a linear transformation $Y=3+2\cdot X$, where $X\sim \mathcal U \{1,6\}$. The expected value and the variance of X are $\mathbb E(X)=3.5$ and $Var(X)= \frac{35}{12}$ Let's see how $Y$ is distributed.

It is uniform distributed as well as $Y\sim \mathcal U \{5,15\}$. And the expected value and the variance can be calculated by using the rules at linear transformated variables.

$$\mathbb E(Y)= \mathbb E(3+2\cdot X)=3+2\cdot \mathbb E(X)=10$$

$$Var(Y)=Var(3+2\cdot X)=2^2\cdot Var(X)=4\cdot \frac{35}{12}=\frac{35}{3}=11\frac23$$

Answer 3

I use this all the time in psychometrics (statistics of tests). Suppose $X$ is the raw score on a 9 item quiz with a mean of $4.5$ and a SD of $2$. Suppose I want to scale the quiz so that is goes from 10-100. I can make each question worth ten points and award 10 points for writing your name. So now $Y=10X+10$. Because the expectation and variance of a linear transformation is easily known, we get $E[Y]=10E[X]+10=55$ and $SD(Y)=10SD(X)=20$.

If you combine this with @parsid's answer, you get a simple equating method for putting two tests on the same scale. First, use his equation to convert a score on scale X to Z (standard normal) then invert the method to convert the Z-score to scale Y.

Answer 4

As an alternative answer, a distribution which remains invariant under a linear transformation has very useful properties. This is another reason why you see Gaussian random variables so much.

For example, consider the discrete-time linear stochastic system \begin{align} x(k+1) =& Ax(k)+Mv(k),\quad v\in G(0,Q_v),\;x_0\in G(m_0,Q_0)\\ y(k) =& Cx(k) \end{align} with $v$ a Gaussian random variable, $x$ a state vector and $y$ some output. A very famous problem in control theory, where linear transformation are ubiquitous, is to see if the state can be obtained through the output. This resulted in the infamous Kalman filter.

I won't go over an entire derivation, but because of this invariance, the long-term statistical properties of the corresponding states and outputs are relatively easily described, better yet, updated.