Statistical distance as Guessing Probability Let X and Y be random variables with probability distributions PX and PY , respectively (i.e., Pr[X = x] = PX(x) and Pr[Y = y] = PY (y). Consider the following random experiment between a challenger and an adversary: 1. The Challenger chooses a bit b {0, 1} If b=0 then the Challenger samples z from the distribution PX else the Challenges samples z from the distribution PY 2. The Challenger sends z to the Adversary A who outputs b 0 . The adversary wins if b 0 = b. Let ExpGuess A denote the random variable which is 1 if the adversary wins, and 0 otherwise. Prove that for every adversary A: Pr[ExpGuess A = 1] 1 2 + (X, Y )

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Statistical distance as Guessing Probability Let X and Y be random variables with probability distributions Px and Py, respectively (i.e., Pr(X = r] Px(z) and Pr(X y] = Py(y). Consider the following random experiment between a challenger and an adversary: 1, The Challenger chooses a bit b {0,1} If b=0 then the Challenger samples z from the distribution Pr else the Challenges samples z from the distribution Py 2. The Challenger sends z to the Adversary A who outputs . The adversary wins it b'=b. Let ExpCuess denote the random variable which is 1 if the adversary wins, and 0 otherwise. Prove that for every adversary A: Hint: Use Task 2(b). Statistical distance as Guessing Probability Let X and Y be random variables with probability distributions Px and Py, respectively (i.e., Pr(X = r] Px(z) and Pr(X y] = Py(y). Consider the following random experiment between a challenger and an adversary: 1, The Challenger chooses a bit b {0,1} If b=0 then the Challenger samples z from the distribution Pr else the Challenges samples z from the distribution Py 2. The Challenger sends z to the Adversary A who outputs . The adversary wins it b'=b. Let ExpCuess denote the random variable which is 1 if the adversary wins, and 0 otherwise. Prove that for every adversary A: Hint: Use Task 2(b)