Questions Pool

1. In class we defined the statistical (also known as variational) distance between two distributions A and B taking values in U to be |A− B| = maxT⊆U |Pr[A ∈ T ]− Pr[B ∈ T ]|. A× B denotes the distribution on (a, b) obtained by independently picking a according to A and b according to B. (A,B) denotes some random variable on U × U with marginal distributions A and B. For f : U → U ′ and A that is distributed over U let f(A) denote the distribution on U ′ obtained by picking a ∼ A and outputting f(a). (a) Prove that |A−B| = 12 ∑ x∈U |A(x)−B(x)|. (b) Prove that |A×B −A′ ×B′| ≤ |A−A′|+ |B −B′|. (c) Show that it is not always the case that |(A,B)− (A′, B′)| ≤ |A−A′|+ |B −B′|. (d) Assume |(B|A = a)−C| ≤ ε for every a ∈ U . Prove that |(A,B)−A×C| ≤ ε. Conclude that if E : {0, 1} → {0, 1} is an extractor for (n, n2 ) oblivious bit-fixing sources with error ε, and f(x) returns n2 bits of x, then |E(X) ◦ f(X)− Um × f(X)| ≤ ε.