Notation for probability of either sign

Question

I have a function $f(k) = \pm 2^k$ with probability $1/2$ of either sign. How would I express this in a cleaner notation? I'm guessing to use the Kronecker delta somehow, but I can't put a finger on it.

Answer 1

You might try $f(k)=2^kX_k$ where $X_k$ is a symmetric Bernoulli random variable (this means that $\mathbb P(X_k=+1)=\mathbb P(X_k=-1)=\frac12$).

Answer 2

How about using 2 cases? \begin{align} f(k)&=\cases{+2^k\qquad w.p.\;1/2\\-2^k\qquad w.p.\;1/2} \qquad k\in \text{Whatever} \end{align}