Statistics

Here, we will consider a random variable xN(,) and provide some bounds on Prob(X+a) or a>0. (a) Show that Prob(X+a)2aexp(22a2) Hint: Start from the expression for Prob(X+a) as an integral and use that for x+a, ex2/22(x)ex2/22/a (b) Show that for any s>0 Prob(X+a)exp(2s22as) Hint: Try to use the Markov's inequality on the random variable es(X). (c) Using the result of (b), show that Prob(X+a)exp(22a2) Here, we will consider a random variable xN(,) and provide some bounds on Prob(X+a) or a>0. (a) Show that Prob(X+a)2aexp(22a2) Hint: Start from the expression for Prob(X+a) as an integral and use that for x+a, ex2/22(x)ex2/22/a (b) Show that for any s>0 Prob(X+a)exp(2s22as) Hint: Try to use the Markov's inequality on the random variable es(X). (c) Using the result of (b), show that Prob(X+a)exp(22a2)