I have this task and would love some feedback on whether I'm going in the right direction:
Suppose the random variable $X$ has distribution given by probability mass function
$$ \begin{array}{c|lcr} x & \text{-2} & \text{0} & \text{2} & \text{4}\\ \hline P(X=x) & 0.2 & 0.5 & 0.1 & 0.2 \\ \end{array} $$
calculate the probability mass function for random variable $Y = 1/2(X-1)^2$.
After reading up on this, this is my answer: we get values $4.5,\ 0.5,\ 0.5$ and $4.5$ for $Y$ and this is the final pmf: $$ \begin{array}{c|lcr} y & \text{0.5} & \text{4.5}\\ \hline P(Y=y) & 0.4 & 0.6 \\ \end{array} $$
It kind of baffles me that the probabilities should be the same for the changed variable, but this is what I heard. Am I correctly assuming that I have to sum up the probabilities if the $Y$ has two same values twice? ($4.5$ and $0.5$). I hope this is the right place to post - I haven't found similar examples so far but I'd love to know how to solve tasks like these. If this is not the right place to post this, please redirect me! Thank you.