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
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa86feb48-aed4-4133-a748-653f8a12a813%2F2201f26b-53f7-49c0-9571-01166e4ce706%2Fire1t9_processed.jpeg&w=3840&q=75)
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
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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Write the given information
VIEWStep 2: What can you conclude about P(Y> ε^2), for any nonzero number ε, using Markov's inequality
VIEWStep 3: Prove that {Xn} converges to 0 in the sense of mean square convergence
VIEWStep 4: Show that Xn converges to 0 in probability using part a in part b
VIEWSolution
VIEW
Step by step
Solved in 5 steps with 40 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman