Probability notation

Probability notation

tau-abm 09 Nov 2019 13:32

Hi,

I want to make sure i'm aligned with the probability notations in the course.

The location of the (random) variables differ between some parts of the scribes/homework.
Is there a difference between the following:

(1)

\begin{align} Pr[A(x) = b | \begin{array}{l} b \gets \{0, 1\} \\ x \gets X_b \end{array}] \end{align}

(2)

\begin{align} \underset{b}{Pr}[A(X_b) = b] \end{align}

Thanks!

Re: Probability notation

nbitansky 09 Nov 2019 15:35

I would interpret them as the same. When using the second I would typically write $\Pr_{b,X_b} [A(X_b) = b]$. Note that we often identify distributions and random variables that are distributed according to a distribution. For instance, if we say that $U_n$ is the uniform distribution on $n$-bit strings we may write $\Pr[U_n = 0^n] = 2^{-n}$, treating it as a r.v., or we can write $$\Pr[u = 0^n | u\gets U_n] = 2^{-n}$, treating it as a distribution. We mean the same thing (and there's really only one reasonable way to interpret all of these).