So, I was going through Pattern Recognition and Machine learning by Bishop, page - 18, and I came across probability densities where I read about the transformation of a random variable. I couldn't understand what it actually means? For example, let's take a die.
Let the outcome of a die roll be represented by the random variable $X$. So, the probability distribution is as follows: $P(X=1)=1/6, P(X=2)=1/6, P(X=3)=1/6, P(X=4)=1/6, P(X=5)=1/6, P(X=6)=1/6$
Now if I make a non-linear transformation of $X$ say, ${{X}^{2}}$ what does this actually mean? From the definition of probability distribution, I think, we can say that the following is the probability distribution of the random variable ${{X}^{2}}$. $P({{X}^{2}}=1)=1/2, P({{X}^{2}}=4)=1/2, P({{X}^{2}}=9)=0, P({{X}^{2}}=16)=0, P({{X}^{2}}=25)=0, P({{X}^{2}}=36)=0$ But how do I interpret ${{X}^{2}}$ w.r.t. a die roll? Also, why would I do such a thing?