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4. a. Let Y be a positive random variable such that E[Y] < where N 2 2. Using Markov's inequality, what can you conclude about P(Y> e), for any nonzero number e? Markov's Inequality If X is any nonnegative random variable, then P(X≥ a) ≤ for any a > 0. b. Suppose (X). n = 1,2,3.... is a sequence of random variables such that E[X] < 1/N for any integer N. Prove that (X] converges to 0 in the sense of mean square convergence. (Following is the definition of convergence in mean square to 0) Convergence in Mean Let r21 be a fixed number. A sequence of random variables X₁, X₂. X. converges in the rth mean or in the L norm to by X. . zero shown lim B (IX.-01)-0. If -2, it is called the mean-square convergence, and it is shown by x. 10

4. a. Let Y be a positive random variable such that E[Y] < where N 2 2. Using Markov's inequality, what can you conclude about P(Y> e), for any nonzero number e? Markov's Inequality If X is any nonnegative random variable, then P(X≥ a) ≤ for any a > 0. b. Suppose (X). n = 1,2,3.... is a sequence of random variables such that E[X] < 1/N for any integer N. Prove that (X] converges to 0 in the sense of mean square convergence. (Following is the definition of convergence in mean square to 0) Convergence in Mean Let r21 be a fixed number. A sequence of random variables X₁, X₂. X. converges in the rth mean or in the L norm to by X. . zero shown lim B (IX.-01)-0. If -2, it is called the mean-square convergence, and it is shown by x. 10

MATLAB: An Introduction with Applications
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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5:49 Page of 2 central mmit theorem to na the probadmity that there will be more than 100 accidents in a certam month. Assume that there are 30 days in a month. If X is any nonnegative random variable, then E 4. a. Let Y be a positive random variable such that E[Y]<, where N 22. Using Markov's inequality, what can you conclude about P(Y> e2), for any nonzero number ? Pll 1 b. Suppose (X). n = 1,2,3,... is a sequence of random variables such that E[X] < 1/N for any integer N. Prove that (Xn) converges to 0 in the sense of mean square convergence. (Following is the definition of convergence in mean square to 0) Convergence in Mean Let r 21 be a fixed number. A sequence of random variables X₁, X₂, X3... converges in the rth mean or in the L norm to zero by Xif shown B L lim E(X.-01) - ⁰. If r-2, it is called the mean-square convergence, and it is shown by x₂ O m.a $ P(X≥ a) ≤EX, for any a > 0. R Markov's Inequality Today 5:40 PM Convergence of RV Pa d % 5 T de A 6 9 Y 3 & 7 U PrtScn 8 Edit i Home ( 44 & Near record 9 End O PgUp In this ariables s 15a56 a in your and S that are Explaie v I just wain your an NVERGEN Por
Transcribed Image Text:5:49 Page of 2 central mmit theorem to na the probadmity that there will be more than 100 accidents in a certam month. Assume that there are 30 days in a month. If X is any nonnegative random variable, then E 4. a. Let Y be a positive random variable such that E[Y]<, where N 22. Using Markov's inequality, what can you conclude about P(Y> e2), for any nonzero number ? Pll 1 b. Suppose (X). n = 1,2,3,... is a sequence of random variables such that E[X] < 1/N for any integer N. Prove that (Xn) converges to 0 in the sense of mean square convergence. (Following is the definition of convergence in mean square to 0) Convergence in Mean Let r 21 be a fixed number. A sequence of random variables X₁, X₂, X3... converges in the rth mean or in the L norm to zero by Xif shown B L lim E(X.-01) - ⁰. If r-2, it is called the mean-square convergence, and it is shown by x₂ O m.a $ P(X≥ a) ≤EX, for any a > 0. R Markov's Inequality Today 5:40 PM Convergence of RV Pa d % 5 T de A 6 9 Y 3 & 7 U PrtScn 8 Edit i Home ( 44 & Near record 9 End O PgUp In this ariables s 15a56 a in your and S that are Explaie v I just wain your an NVERGEN Por
rch 5:49 E M c. Let (X) be as in part b. Use part a to show that X, converges to 0 in probability. Following is the definition of convergence in probability to zero: Convergence in Probability A sequence of random variables X₁, X₂, X3. converges in probability to zero, shown by X, O, if D lim P(|X, -012)-0, for all > 0. B n DII Convergence of RV Page 1 Hint: Use part a. with Y= X to get an expression for P(IXI2e), and show that the limit is 0 when n → co. $ 4 R d % 5 T Today 5:40 PM 3 49 A 6 Y Se F7 & 7 U PrtScn 8 (i) Edit Home I ( 9 End 44 & Near record In the ariables su this a 15 56 I your an S that are ? Explain y I just wain your an NVERGEN 0 D PgUp P EP
Transcribed Image Text:rch 5:49 E M c. Let (X) be as in part b. Use part a to show that X, converges to 0 in probability. Following is the definition of convergence in probability to zero: Convergence in Probability A sequence of random variables X₁, X₂, X3. converges in probability to zero, shown by X, O, if D lim P(|X, -012)-0, for all > 0. B n DII Convergence of RV Page 1 Hint: Use part a. with Y= X to get an expression for P(IXI2e), and show that the limit is 0 when n → co. $ 4 R d % 5 T Today 5:40 PM 3 49 A 6 Y Se F7 & 7 U PrtScn 8 (i) Edit Home I ( 9 End 44 & Near record In the ariables su this a 15 56 I your an S that are ? Explain y I just wain your an NVERGEN 0 D PgUp P EP
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