Exercise B.11. Let X be a zero-mean, 1-sub-Gaussian random variable. Show that there exists a universal...

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Exercise B.11. Let X be a zero-mean, 1-sub-Gaussian random variable. Show that there exists a universal constant c> 0 such that for all p 1, ||X||L, := (E|X|P)/ < cp. Hint: You will need to utilize the following facts (which you can use without proof): (a) For a non-negative random variable X, E[X] = P(X > t) dt. (b) For all s 1/2, F(s) 3s, where (s) := ts-et dt is the Gamma function. (c) The quantity supp1 p/p < . Exercise B.11. Let X be a zero-mean, 1-sub-Gaussian random variable. Show that there exists a universal constant c> 0 such that for all p 1, ||X||L, := (E|X|P)/ < cp. Hint: You will need to utilize the following facts (which you can use without proof): (a) For a non-negative random variable X, E[X] = P(X > t) dt. (b) For all s 1/2, F(s) 3s, where (s) := ts-et dt is the Gamma function. (c) The quantity supp1 p/p < .