Let {Xn} be a sequence of random variables such that the following limits (in the regular point-wise sense) hold: and lim E[X₂] = c for some constant c, n→∞ lim Var[X₂] = 0. n→∞ Show that Xn converges to c in the mean-square sense, that is, lim E[(Xn-c)²] = 0. n→∞ Hint: write E[(X - c)²] in a way that it involves Var[Xn].
Let {Xn} be a sequence of random variables such that the following limits (in the regular point-wise sense) hold: and lim E[X₂] = c for some constant c, n→∞ lim Var[X₂] = 0. n→∞ Show that Xn converges to c in the mean-square sense, that is, lim E[(Xn-c)²] = 0. n→∞ Hint: write E[(X - c)²] in a way that it involves Var[Xn].
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Let \(\{X_n\}\) be a sequence of random variables such that the following limits (in the regular point-wise sense) hold:
\[
\lim_{n \to \infty} \mathbb{E}[X_n] = c \quad \text{for some constant } c,
\]
and
\[
\lim_{n \to \infty} \text{Var}[X_n] = 0.
\]
Show that \(X_n\) converges to \(c\) in the mean-square sense, that is,
\[
\lim_{n \to \infty} \mathbb{E}[(X_n - c)^2] = 0.
\]
Hint: write \(\mathbb{E}[(X_n - c)^2]\) in a way that it involves \(\text{Var}[X_n]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ad3190e-c519-4062-ad62-d2f3b1999b5f%2F6a16c18d-1218-4eb9-87d6-46d0169b8da9%2Fqt7fkz_processed.png&w=3840&q=75)
Transcribed Image Text:Let \(\{X_n\}\) be a sequence of random variables such that the following limits (in the regular point-wise sense) hold:
\[
\lim_{n \to \infty} \mathbb{E}[X_n] = c \quad \text{for some constant } c,
\]
and
\[
\lim_{n \to \infty} \text{Var}[X_n] = 0.
\]
Show that \(X_n\) converges to \(c\) in the mean-square sense, that is,
\[
\lim_{n \to \infty} \mathbb{E}[(X_n - c)^2] = 0.
\]
Hint: write \(\mathbb{E}[(X_n - c)^2]\) in a way that it involves \(\text{Var}[X_n]\).
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is a sequence of random variables.
; for some constant .
Need to prove: .
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