I am George P. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, Malacca, Malaysia. I have been helping students with their homework for the past 8 years. I solve assignments related to Stochastic Processes.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
I am Martin J. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
I am Martin J. I am a Stochastic Processes Assignment Expert at excelhomeworkhelp.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
My talk in the MCQMC Conference 2016, Stanford University. The talk is about Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks.
I am Martin J. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
I am Martin J. I am a Stochastic Processes Assignment Expert at excelhomeworkhelp.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
My talk in the MCQMC Conference 2016, Stanford University. The talk is about Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks.
The Excel Homework Help team comprises highly experienced academic writers who possess the expertise to craft original and unique content for all sorts of Excel homework. Our professionals will meticulously prepare a well-structured academic paper that meets all your requirements without any trace of plagiarism. You can easily access various types of Excel homework writing services from the convenience of your home.
Reach out to our team via: -
Website: https://www.excelhomeworkhelp.com/
Email: support@excelhomeworkhelp.com
Call/WhatsApp: +1(315)557–6473
I am Arcady N. I am a Computer Network Assignments Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, City University, London. I have been helping students with their assignments for the past 10 years. I solve assignments related to the Computer Network.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with the Computer Network Assignments.
I am Martina J. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Maryland. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
I am Falid B. I am a Mathematical Statistics Assignment Help Expert at statisticshomeworkhelper.com. I hold a Master's in Statistics, from George Town, Malaysia.I have been helping students with their assignment for the past 6 years. I solve assignments related to Mathematical Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.You can also call on +1 678 648 4277 for any assistance with Mathematical Statistics Assignment.
I am Falid B. I am a Mathematical Statistics Assignment Expert at excelhomeworkhelp.com. I hold a Master's in Statistics, from George Town, Malaysia. I have been helping students with their assignments for the past 6 years. I solved an assignment related to Mathematical Statistics.
Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Mathematical Statistics Assignment.
A set of notes prepared for an introductory machine learning course, assuming very limited linear algebra background, because all linear algebra operations are fully written out. These notes go into thorough derivations of the generalized linear regression formulation, demonstrating how to write it out in matrix form.
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
I am Isaac M. I am a Computer Network Assignment Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, Glasgow University, UK. I have been helping students with their assignments for the past 8 years. I solve assignments related to the Computer Network.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with the Computer Network Assignment.
Implicit schemes are needed in order to have fast runtime in wave models. Parallelization using the Message Passing Interface are needed in order to run on computers with thousands of processors. Implicit schemes rely on preconditioner in order for the iterative schemes to converge fast. Thus we need fast preconditioners and we present those here.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
Need help with your statistics assignments? Our website offers top-notch statistics assignment help to students at all academic levels. Whether you're struggling with data analysis, hypothesis testing, or any other statistical concept, our experienced team of experts is here to assist you. We provide accurate solutions, clear explanations, and timely delivery to ensure your success. Our user-friendly platform makes it easy to submit your assignments and communicate with our dedicated professionals. Don't let statistics stress you out—visit our website today and let us handle your assignments with precision and expertise. Get the grades you deserve with our reliable statistics assignment help service!
Looking for professional assistance with your statistics assignments? Our website provides reliable “Statistics Assignment Help” services to help you with your assignment. Our experienced team of experts can provide personalized support and guidance, ensuring you understand the subject matter and achieve your academic goals. Contact us today to learn more about our services and how we can help you succeed.
🌐Website:- www.statisticsassignmenthelp.com
📧Email:- support@statisticsassignmenthelp.com
📲Call/WhatsApp:- +1(315)557–6437
The Excel Homework Help team comprises highly experienced academic writers who possess the expertise to craft original and unique content for all sorts of Excel homework. Our professionals will meticulously prepare a well-structured academic paper that meets all your requirements without any trace of plagiarism. You can easily access various types of Excel homework writing services from the convenience of your home.
Reach out to our team via: -
Website: https://www.excelhomeworkhelp.com/
Email: support@excelhomeworkhelp.com
Call/WhatsApp: +1(315)557–6473
I am Arcady N. I am a Computer Network Assignments Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, City University, London. I have been helping students with their assignments for the past 10 years. I solve assignments related to the Computer Network.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with the Computer Network Assignments.
I am Martina J. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Maryland. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
I am Falid B. I am a Mathematical Statistics Assignment Help Expert at statisticshomeworkhelper.com. I hold a Master's in Statistics, from George Town, Malaysia.I have been helping students with their assignment for the past 6 years. I solve assignments related to Mathematical Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.You can also call on +1 678 648 4277 for any assistance with Mathematical Statistics Assignment.
I am Falid B. I am a Mathematical Statistics Assignment Expert at excelhomeworkhelp.com. I hold a Master's in Statistics, from George Town, Malaysia. I have been helping students with their assignments for the past 6 years. I solved an assignment related to Mathematical Statistics.
Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Mathematical Statistics Assignment.
A set of notes prepared for an introductory machine learning course, assuming very limited linear algebra background, because all linear algebra operations are fully written out. These notes go into thorough derivations of the generalized linear regression formulation, demonstrating how to write it out in matrix form.
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
I am Isaac M. I am a Computer Network Assignment Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, Glasgow University, UK. I have been helping students with their assignments for the past 8 years. I solve assignments related to the Computer Network.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with the Computer Network Assignment.
Implicit schemes are needed in order to have fast runtime in wave models. Parallelization using the Message Passing Interface are needed in order to run on computers with thousands of processors. Implicit schemes rely on preconditioner in order for the iterative schemes to converge fast. Thus we need fast preconditioners and we present those here.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
Similar to Stochastic Processes Assignment Help (20)
Need help with your statistics assignments? Our website offers top-notch statistics assignment help to students at all academic levels. Whether you're struggling with data analysis, hypothesis testing, or any other statistical concept, our experienced team of experts is here to assist you. We provide accurate solutions, clear explanations, and timely delivery to ensure your success. Our user-friendly platform makes it easy to submit your assignments and communicate with our dedicated professionals. Don't let statistics stress you out—visit our website today and let us handle your assignments with precision and expertise. Get the grades you deserve with our reliable statistics assignment help service!
Looking for professional assistance with your statistics assignments? Our website provides reliable “Statistics Assignment Help” services to help you with your assignment. Our experienced team of experts can provide personalized support and guidance, ensuring you understand the subject matter and achieve your academic goals. Contact us today to learn more about our services and how we can help you succeed.
🌐Website:- www.statisticsassignmenthelp.com
📧Email:- support@statisticsassignmenthelp.com
📲Call/WhatsApp:- +1(315)557–6437
Are you struggling with your statistics assignment and looking for reliable help? Don't worry! Our website provides professional assignment help that guarantees accurate and timely solutions. Pay for statistics assignment and get a team of experienced experts who will work on your assignment and deliver quality results. We provide assistance with various topics, including hypothesis testing, regression analysis, probability theory, and much more.
🌐Website:- www.statisticsassignmenthelp.com
📧Email:- support@statisticsassignmenthelp.com
📲Call/WhatsApp:- +1(315)557–6437
I am Abigail Taylor. I am a statistics Assignment Help Expert at statisticsassignmenthelp.com. I hold a Master's in statistics, from the California Institute of Technology, USA. I have been helping students with their Probability assignments for the past 6 years.
Visit statisticsassignmenthelp.com or email support@statisticsassignmenthelp.com.
You can also call +1 (315) 557-6473.
I am Martha Anderson. I love exploring new topics. Academic writing seemed an exciting option for me. After working for many years with statisticsassignmenthelp.com as a statistics Assignment Help Expert, I have assisted many students with their Data Analysis assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided.
My name is Penny Forell. I am associated with statisticsassignmenthelp.com as a Statistics Assignment help expert for the past 6 years and have been helping students with their R Programming assignments. I have a Master’s in statistics, from the University of Chicago, United States.
I am Ann Brenda Mathews. Currently associated with statisticsassignmenthelp.com as a statistics assignment help Expert. After completing my Master’s in Statistics, at the University of Liverpool, I was in search of an opportunity that would expand my area of knowledge hence I decided to help students with their assignments. I have written several Hypothesis Assignments to help students overcome numerous difficulties.
The Data of an Observational Study Designed to Compare the Effectiveness of a...Statistics Assignment Help
I am Melissa Maribeth. I love exploring new topics. Academic writing seemed an exciting option for me. After working for many years with statisticsassignmenthelp.com, I have assisted many students with their statistics assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I acquired my bachelor's from Princeton University United States.
My name is Mathew Olson. I am associated with statisticsassignmenthelp.com for the past 5 years and have been helping statistics students with their T-Test and ANOVA using SPSS Assignments. I have a Master’s in Statistics, from Florida State University.
I am Sarah Reynolds, Currently associated with statisticsassignmenthelp.com as a Statistics assignment helper. After completing my master's from the University of London, UK. I was in search of an opportunity that would expand my area of knowledge hence I decided to help students with their assignments. I have written several statistics assignments to date to help students overcome numerous difficulties they face in Linear Regression Analysis assignments.
I am Bethany David. Currently associated with statisticsassignmenthelp.com as a statistics assignment helper. After completing my master's from the University of London, UK. I was in search of an opportunity that would expand my area of knowledge hence I decided to help students with their assignments. I have written several statistics assignments to date to help students overcome numerous difficulties they face in State assignments.
My name is Rose Tom. I am associated with statisticsassignmenthelp.com for the past 8 years and have been helping statistics students with their MyStataLab assignments.
I have a master's in Professional Statistics from Princeton University, USA.
I am Jacinta Lawrence Currently associated with statisticsassignmenthelp.com as a Probability and Statistics assignment helper. After completing my master's from Princeton University, USA. I was in search of an opportunity that would expand my area of knowledge hence I decided to help students with their assignments. I have written several statistics assignments to date to help students overcome numerous difficulties they face.
I am Jason B. I am a Mathematical Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from Princeton University, USA. I have been helping students with their homework for the past 9 years. I solve assignments related to Mathematical Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Mathematical Statistics Assignments.
I am Thanasis F. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from Harvard University. I have been helping students with their homework for the past 9 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I am Parton R. I am a Statistics Coursework Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from The University of Edinburgh, UK. I have been helping students with their assignments for the past 8 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I am Leonard K. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from, the University of Arkansas, USA. I have been helping students with their homework for the past 9 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I am Frank P. I am a Statistics Coursework Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from Malacca, Malaysia. I have been helping students with their assignments for the past 10 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I am Craig D. I am a Probabilistic Assignment Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from Malacca, Malaysia.
I have been helping students with their assignments for the past 6 years. I solve assignments related to Probabilistic Systems Analysis.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Probabilistic Systems Analysis Assignments.
I am Baddie K. I am a Probabilistic Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from The University of Queensland.
I have been helping students with their homework for the past 9 years. I solve assignments related to Probabilistic Systems Analysis.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Probabilistic Systems Analysis Assignments.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Delivering Micro-Credentials in Technical and Vocational Education and TrainingAG2 Design
Explore how micro-credentials are transforming Technical and Vocational Education and Training (TVET) with this comprehensive slide deck. Discover what micro-credentials are, their importance in TVET, the advantages they offer, and the insights from industry experts. Additionally, learn about the top software applications available for creating and managing micro-credentials. This presentation also includes valuable resources and a discussion on the future of these specialised certifications.
For more detailed information on delivering micro-credentials in TVET, visit this https://tvettrainer.com/delivering-micro-credentials-in-tvet/
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
1. For any help regarding Stochastic Processes Assignment Help
Visit : https://www.statisticsassignmenthelp.com/ ,
Email - info@statisticsassignmenthelp.com or
Call us at - +1 678 648 4277
2. Solution to Exercise 2.28:
The purpose of this problem is to illustrate that for an arrival process with
independent but not identically distributed interarrival intervals, X1,X2,...,, the
number of arrivals N(t) in the interval (0,t] can be a defective rv. In other words,
the ‘counting process’ is not a stochastic process according to our definitions. This
illustrates that it is necessary to prove that the counting rv’s for a renewal
process are actually rv’s.
a) Let the distribution function of the ith interarrival interval for an arrival
process be FXi (x i) = 1 − exp(−α− xi) for some fixed α ∈ (0, 1). Let Sn = X1 + ··· Xn
and show that
b) Sketch a ‘reasonable’ sample function for N(t).
c) Find σ2 Sn .
d) Use the Chebyshev inequality on Pr Sn ≥ t to find an upper bound on Pr N(t)
≤ n that is smaller than 1 for all n and for large enough t. Use this to show
that N(t) is defective for large enough t.
SOLUTION: a) Each Xi is an exponential rv, of rate αi, so E[Xi] = αi . Thus
statisticsassignmenthelp.com
3. Recalling that 1 + α + α2 + ··· = 1/(1 − α),
In other words, not only is E[Xi] decaying to 0 geometrically with increasing i,
but E[Sn] is upper bounded, for all n, by α/(1 − α).
b) Since the expected interarrival times are decaying geometrically and the
expected arrival epochs are bounded for all n, it is reasonable for a sample path
to have the following shape:
Note that the question here is not precise (there are obviously many sample paths, and
which are ‘reasonable’ is a matter of interpretation). The reason for drawing such
sketches is to acquire understanding to guide the solution to the following parts of the
problem. c) Since Xi is exponential, σ2= α2i X. Since the Xi i are independent,
statisticsassignmenthelp.com
4. d) The figure suggests (but does not prove) that for typical sample functions (and in
particular for a set of sample functions of nonzero probability), N(t) goes to infinity for
finite values of t. If the probability that N(t) ≤ n (for a given t) is bounded, independent
of n, by a number strictly less than 1, then that N(t) is a defective rv rather than a true
rv.
The problem suggests using the Chebyshev bound to prove this, but the Markov
bound is both simpler and also shows that N(t) is defective for smaller values of t than
established by Chebyshev. Thus we first use Markov and then use Chebyshev. By
Markov,
where we have used Eq. 2.3 in the text. Since this is valid for all n, it is also valid for Pr
N(t) ≤ n. For any t > α/(1 − α), we see that α< 1. Thus N(t) is defective for t(1−α)
anysuch t, i.e., for any t greater than limn→∞ ESn.
Now to use Chebyshev, note that
statisticsassignmenthelp.com
5. Now σ2 Sis increasing in n to the limit σ2 = α2/(1 α2). Also S¯n− n is increasing to the
∞limit S¯= α/(1 α). Thus (t S¯)−2) is also increasing in n, and
Thus Pr{N(t) ≥ n} is bounded for all n by a number less than 1 (and thus N(t) is
defective) if t ¯− Sσ∞ >, i.e., if t> α σ∞1α+.− ∞
Note that this result is weaker (and considerably more complicated) than that using
the Markov bound. Actually, by working a good deal harder, it can be shown that N(t)
is defective for all t> 0. The outline of the argument is as follows: for any given t, Thus
Pr{N(t) ≥ n} is bounded for all n by a number less than 1 (and thus N(t) is defective) if
t ¯− Sσ∞ >, i.e., if t> α σ∞1α+.− ∞Note that this result is weaker (and considerably
more complicated) than that using the Markov bound. Actually, by working a good
deal harder, it can be shown that N(t) is defective for all t> 0. The outline of the
argument is as follows: for any given t,
statisticsassignmenthelp.com
6. Solution to Exercise 3.2:
a) Based on the definition of the recurrent states, if a state is accessible from all
the states that are accessible from it, (i.e., if for all i2 that i1 → i2, we also
have i2 → i1), the state i1 is called recurrent. So if a state is not recurrent
(i.e., it is transient), there should exist at least one state i2 that is accessible
from i1 (i1 → i2) but i1 is not accessible from it (i2 i1).
b) The same reasoning as in part (a) shows that if i2 is transient, there should
exist a state i3 where i2 → i3 but i3 i2. Additionally, we know that i1 → i2
and i2 → i3 imply that i1 → i3.
We know that i1 → i2. Now if i3 → i1, then i3 → i2 which is against what
we said about i3. Then i1 should not be accessible from i3. (i.e., i3 i1)
c) The same reasoning as in part (a) shows that if ik is transient, there should
exist a state ik+1 for which ik → ik+1 but ik+1 ik. Now if for any 1 ≤ j ≤ k,
ik+1 → j, then we also have ik+1 → ik which is against the assumption about
ik.
d) Each state ik constructed in part c) is distinct from states i1,...,ik 1, since there
is − a path from ij to ik but not from ik to ij . Since there are only J states,
the process in part c) must terminate with some k ≤ J, with means that ik
isn’t transient, i.e., is recurrent.
statisticsassignmenthelp.com
7. Solution to Exercise 3.3:
In exercise 3.2 it is proved that each Markov chain has at least one recurrent
state which induces a recurrent class of states. State 1 is accessible from all
the states. So it is also accessible from the states of the recurrent class in this
Markov chain. According to the definition of recurrent class, all the states that
are accessible from the states of a recurrent class should have access to the
states of this class, meaning that state 1 should communi cate to all of the
classes of recurrent states. If two recurrent states communicate with one state,
they should also communicate with each other, i.e., they belong to a single
recurrent class. Hence, there should exist at most one recurrent class of states
in this Markov chain.
Problem 4:
The desired graph is shown below. It should be clear that this construction can
be used for any number of states M ≥ 3. Starting in state 1, there is only one
possible state (first 2, then 3, up to J) that the chain can be for the first J − 1
steps. On the Jth step, there are two possible states (1 and 2). For each step
from M to 2M − 2, there are two possible states (first 1,2, then 2,3, up to M
− 1,M. Similarly, from steps 2(M − 1) + 1 to 3(M − 1), there are 3 possible
states (first 1, 2, 3, then 2, 3, 4, etc.). This continues up to (M − 1)2, which is
the last step at which there are M 1 possible states, and then from ( − + 1
on, all states are possible at each transition.
statisticsassignmenthelp.com
8. This pattern should be clear from your tabulation for M = 4. The desired graph
is shown below. It should be clear that this construction exists for all J ≥ 3.
Any walk from node J back to node J must be of length iJ + k(J − 1) for some
i ≥ 1 and some k ≥ 1. Note that iJ + k(J − 1) = i +(i + k)(J − 1). Trying to set this
equal to (J − 1)2 yields i + (i + k − J + 1)(J − 1) = 0. This implies that i is a
multiple J − 1, which makes iJ > (J − 1)2, so there is no walk from node J back
to itself in (J − 1)2 steps.
Solution to Exercise 3.8:
[P ] is doubly stochastic if both [P ] and [P ]T are stochastic. This is just a
different way of saying that all the rows and all the columns of [P ] must sum
to one (and all entries are non-negative). Since [P ] is stochastic, it has e (a
column vector) as a right eigenvector with the associated eigenvalue λ = 1,
i.e., [P ]e = e. Since [ P ]T is stochastic also, [P ]Te = e. Taking transposes of
both sides we get eT [P ] = eT , so eT is a left eigenvector of [P ] asso ciated
with λ = 1. Theorem 3.3.1 guarantees that this left eigenvector is unique
within a scale factor, so scaling eT to become a probability vector, π = [1/M,
1/M/, ··· , 1/M]. statisticsassignmenthelp.com
9. Solution to Exercise 3.9: a) K The steady state distribution of the Markov chain.
is a probability vector π for which j=1 πj = 1 and for each 1 ≤ j ≤ K, πj = i=1
Pij πi. In this Markov chain, We can have the following equations:
From the first equations we can have:
Using the second equation and π1 = pπ0
The last equation says = pππ. ρ = pk1 Calling, 1for 0 j k 1 we should −−p k−21−p
≤≤−have:
statisticsassignmenthelp.com
12. As you observe ρ can be considered as a measure of flow among the states in
the Markov chain. When ρ < 1, the flow is towards left and the state 0 has the
highest probability. When ρ > 1, the flow is towards right and the state k−1 has
the highest probability. When ρ = 1, there is no flow in any specific direction
and the steady state distribution is uniform.
c) If ρ = 1, we have:
So if ρ< 1, as k →∞, π0 → 1 − ρ and πk−1 → 0. And if ρ> 1, as k →∞, π0 → 0 and
πk−1 → 1 − 1/ρ. For ρ = 1, the steady state distribution is uniform and π0 = πk−1
=1/k → 0. For ρ< 1, the PMF drops off geometrically with ρ, the process returns to
the zero state often and rarely wanders off to the higher states. Taking the limit,
limk π=→∞0 (1 − ρ), limk πk1 = 0. For ρ = 1, all probabilities are equal, and limk π0
= 0, →∞−→∞limk πk1 = 0. For ρ> 1, the probabilities increase with i and as k gets
larger, →∞−it is increasingly unlikely to be in state 0. Taking the limits, limk→∞ π0 =
(1 − ρ), limk→∞ πk−1 = 0.
statisticsassignmenthelp.com
13. Solution to Exercise 3.10:
a) The transition matrix of the first graph is:
The steady state distribution will satisfy the following equations:
Solving these set of equations, we get:
This was almost obvious considering the symmetry of the graph and equations.
statisticsassignmenthelp.com
14. b)
Let’s define q =2p(1 − p). Then the corresponding Markov chain to [P1]2 is:You can
observe that as the original Markov chain is periodic with period d =2, [P ]d partitions
the Markov chain into d disjoint recurrent classes.
Since, [P ]2 has two recurrent classes, there are two independent steady state
distributions for this Markov chain.
statisticsassignmenthelp.com
15. Any linear combination of these steady state distributions is also a steady state
distribution of this Markov chain. c) If the initial state of the Markov chain is 1 or 3,
i.e., X0 ∈{1, 3}, the steady state distribution at time 2n as n →∞, is [1/2, 0, 1/2, 0]. If
the initial state of the Markov chain is 2 or 4, i.e., X0 ∈{2, 4}, the steady state
distribution at time 2n as n →∞, is [0, 1/2, 0, 1/2]. Thus we will have:
statisticsassignmenthelp.com