Notes in Week 2

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Published	09/18/2023	Toss a fair coin n times: what is the probability that we get at least 75% heads? Obtain the exact expression (which needs to be evaluated numerically…
Published	09/23/2023	Assume you are given a randomized algorithm for solving some decision problems. The algorithm returns the correct answer with probability \(1/2+\…
Published	09/18/2023	Let \(Y\) be \(\chi^2\) distributed with \(n\) degrees of freedom, i.e.,\(Y = \sum_{i=1}^n Z_i^2\) where the \…
Published	09/18/2023	Let \(X_1,\dotsc,X_n\) be independent \(d\)-dimensional vectors with Rademacher entries. How large can we choose \(n\) if we …
Published	09/18/2023	Prove that the operator norm of a symmetric \(n\times n\) square matrix can be expressed as in\[\lVert A \rVert = \max_{x \in S^{n-1}} \lver…
Published	09/18/2023	Show that for two matrices \(A, B\) of compatible dimensions \(\lVert A B \rVert \leq \lVert A \rVert \lVert B \rVert\)
Published	09/18/2023	Let \(A\) be a sub-Gaussian vector with parameter 1. Let \(\Sigma\) be a symmetric matrix. Show that \(\Sigma A\) is sub…
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