MATH 475 - Spring 2022 – Homework 2
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MATH 475 - Spring 2022 – Homework 2
Questions
1. Let A e Rm×l and B e Rm×n be matrices. Consider the Frobenius norm minimization problem
min |AC - B|F 9
CeRl ×n
Show that
= AtB .
is a minimizer of this problem, where At is the pseudoinverse of A.
2. If Ⅹ is a random variable, its αumulBnt∶礻↓n↓rBtin礻 |unαtion is CX(θ) = log (E (exp(θⅩ))) 9
(a) Let Ⅹ1. 9 9 9 . Ⅹn be independent random variables and ≠. θ > 0. Using Markov’s inequal- ity, show that
P ╱ i1 Ⅹi > ≠\ < e-θtE ╱exp ╱θ i1 Ⅹi \\ 9
(b) Using (a), show that
P ╱ i1 Ⅹi > ≠\ < exp ╱θ(i)0(f)『 i1 CXi (θ) - θ≠ 、\ .
where CXi(θ) is the cumulant-generating function of Ⅹi .
3. (a) A -↓rnoulli random variable Ⅹ satisfies P(Ⅹ = 1) = P(Ⅹ = -1) = 1/2. For such a random variable show that
CX (θ) < θ2 /29
Hint: cosh(α) < ex2 /2 .
(b) Let Ⅹ1. 9 9 9 . Ⅹn be independent Bernoulli random variables, and a = (α1. 9 9 9 . αn) e R, a 0. Using Question 2 and part (a), show that
P ╱ i1 αiⅩi > ≠\ < exp ╱ - \ 9
for any ≠ > 0.
(c) Show that
P ╱ │i1 αiⅩi │ > ≠\ < 2 exp ╱ - \ 9
2022-03-14