Notes in Week 1

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Status	Last Update	Fields
Published	09/19/2023	Let $X$ be a random variable and $p \in (0,\infty)$. Show that \[E[\lvert X \rvert] = \int_0^\infty p t^{p-1} \mathbb{P}[\lvert X \rvert…
Published	09/19/2023	Deduce Chebyshev's inequality from Markov's inequality.
Published	09/15/2023	Prove that if $X$ is sub-Gaussian with parameter $\sigma$ then $-X$ is also sub-Gaussian with the same parameter.
Published	09/15/2023	Let $X$ be sub-Gaussian with parameter $\sigma$. Use the union bound and sub-Gaussian bound to show that[$]\mathbb{P}[\lvert X - \m…
Published	09/15/2023	Prove that bounded r.v's are sub-Gaussian. Specifically prove that if $X \in [-a,a]$ with probability 1, and $\mathbb{E}[X] = 0$&nb…
Published	09/15/2023	Let $X$ be sub-Gaussian. Prove that $X^2-\mathbb{E}[X^2]$ is sub-exponential. Hint: take the Taylor expansion of \(\mathb…
Published	09/15/2023	Assume that $X$ has zero mean and is sub-Gaussian with parameter $\sigma$. Show that \(\mathbb{E}[\lvert X \rvert^p] \leq 2^{p…
Published	09/15/2023	Let $\phi(x)$ and $\Phi(x)$ be the density and distribution function, respectively, of the standard normal distribution. Show,…
Status	Last Update	Fields

Status	Last Update	Fields
Published	09/19/2023	Let \(X\) be a random variable and \(p \in (0,\infty)\). Show that \[E[\lvert X \rvert] = \int_0^\infty p t^{p-1} \mathbb{P}[\lvert X \rvert…
Published	09/19/2023	Deduce Chebyshev's inequality from Markov's inequality.
Published	09/15/2023	Prove that if \(X\) is sub-Gaussian with parameter \(\sigma\) then \(-X\) is also sub-Gaussian with the same parameter.
Published	09/15/2023	Let \(X\) be sub-Gaussian with parameter \(\sigma\). Use the union bound and sub-Gaussian bound to show that[$]\mathbb{P}[\lvert X - \m…
Published	09/15/2023	Prove that bounded r.v's are sub-Gaussian. Specifically prove that if \(X \in [-a,a]\) with probability 1, and \(\mathbb{E}[X] = 0\)&nb…
Published	09/15/2023	Let \(X\) be sub-Gaussian. Prove that \(X^2-\mathbb{E}[X^2]\) is sub-exponential. Hint: take the Taylor expansion of \(\mathb…
Published	09/15/2023	Assume that \(X\) has zero mean and is sub-Gaussian with parameter \(\sigma\). Show that \(\mathbb{E}[\lvert X \rvert^p] \leq 2^{p…
Published	09/15/2023	Let \(\phi(x)\) and \(\Phi(x)\) be the density and distribution function, respectively, of the standard normal distribution. Show,…
Status	Last Update	Fields