Exercise 1 (Exponential distribution). Let X ~ Exp(1), that is, X has density function p(x) = e¯×1, 1. Calculate the mean and the the second moment of X. You can use the integration by parts formula....

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Extracted text: Exercise 1 (Exponential distribution). Let X ~ Exp(1), that is, X has density function p(x) = e¯×1, 1. Calculate the mean and the the second moment of X. You can use the integration by parts formula. 2. Calculate E[X-1/2]. You can use a Gaussian integral formula. {x20}• → (R, B(R)) which is bounded in L². Exercise 2 (Cantelli's inequality). Consider a random variable X : (2, F, P) Denote by u = B[X] its average and o2 = E[(X – µ)²] its variance. 1. Show that for all A > 0 and u > 0, o2 +u? E[(X – µ + u)²] (A + u)2 P[X - μ > λ] < (а+="" и)?="" 2.="" deduce="" cantelli's="" inequality:="" {="" if="" a=""> 0; > 1- if A < 0.="" pix="" -="" μ="" σ="">