What does it mean when a random variable Y is equal to $2^X$ where X is a discrete random variable?

Question

I'm trying to see what is the ratio of $H(X)$ and $H(Y)$ when $Y=2^X$ .$H$ is the entropy of a random variable.

I have tried using the definition of entropy, but i don't understand what is $2^X$ to continue with this approach.

Answer 1

The entropy of a random variable has nothing to do with the particular values of the variable. Your variable $X$ has certain values $x_i$, which are taken with probabilities $p_i$. The random variable $Y$ then has the values $y_i=2^{x_i}$, which are taken by the same probabilities $p_i$. As $t\mapsto 2^t$ is injective it follows that $$H(Y)=H(X)=-\sum_i p_i\>\log_2 p_i\ .$$