(a) Show that, if X is a real-valued random variable and ?,...

(a) Show that, if X is a real-valued random variable and ?, p > 0, then ?(|X| > ?) ? ?(|X|^p) / ?^p. (b) Let (S_n)_n be a sequence of real-valued random variables such that ?(|S_n|^3) ? n. Prove that ½ S_n ? 0 almost surely, as n ? ?.

Transcript

00:01 Hello students in the given problem we need to show that probability of determinant of x greater than lambda is less than expected value of determinant of x to the power p divided by lambda to the power.

00:14 Here we are going to use markov's inequality which states that for any non -negative random variable y and any constant a greater than 0 we have probability of y greater than or equal to the expected value of y divided by a provided that a is greater than 0.

00:46 Applying markov's inequality to the random variable y is equal to determinant of x to the power p and a constant a equal to lambda we have probability of determinant of x to the power p greater than lambda to the power p less than or equal to expected value of determinant of x to the power p divided by lambda to the power p.

01:13 Since determinant of x greater than lambda is equivalent to determinant of x to the power p greater than or equal to lambda power p.

01:22 We can rewrite the equation as probability of determinant of x greater than lambda is less than or equal to expected value of determinant of x to the power p divided by lambda power p.

01:37 This proves the first part.

01:39 For the second part, we need to consider the sequence sn of a real valued random variable such that expected value of determinant of sn to the power 3 is less than or equal to n.

01:55 Here we need to prove that 1 upon n sn converges to 0 almost surely as n approaches to infinity.

02:05 To prove this, we need to show that for any epsilon greater than 0, the probability of the event of omega determinant of 1 upon n sn omega minus 0 is greater than or equal to epsilon converges to 0 as n approaches to infinity.

02:26 So first we assume that let epsilon greater than 0 be given and now we want to show that probability of determinant of 1 upon n sn minus 0 is greater than or equal to epsilon approaches to 0 as n converges to infinity...

Question

(a) Show that, if X is a real-valued random variable and ?, p > 0, then ?(|X| > ?) ? ?(|X|^p) / ?^p. (b) Let (S_n)_n be a sequence of real-valued random variables such that ?(|S_n|^3) ? n. Prove that ½ S_n ? 0 almost surely, as n ? ?.

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