I followed through the whole series, and this is the BEST Statistics course I have ever taken. Thank you Prof Tsitsiklis for being so clear and concise. I had never imagined a starting statistics course enhancing my understanding so much and giving me so much confidence. This is magic!
I've never come across a more eloquent, immaculate and lucid introduction to random processes before. It's a blessing to have the opportunity to listen to your lecture, Prof. Tsitsiklis. :)
While deriving distribution of time until kth arrival, it appears we have assumed that there will never be two arrivals in succession..! Does the argument change any bit if this assumption doesn't hold..? More so are we really correct to assume this at the first place?
38:38 I did not get what he meant. Y is a random variable indicating the time we have to wait. And for time, isn't it continuous? Then Y should also be a continuous RV. Why did he mention PMF rather than PDF?
Technically the set of all infinite binary strings maps to the any countably infinite set such as the integers or the rational numbers, but not the real numbers.
Not sure about that. Even irrational numbers have binary representations (if you can make it in base 10, you can make it in base 2). I would think the set ({binary representations of all rationals} union {binary representations of all irrationals}) would biject pretty easily to the reals, and would be a subset of {all infinite binary strings}.
There are countable infinity such as integers and uncountable infinity such as real numbers. The Set chapter of mathematics for computer science discusses this difference.
30:30 why is professor saying that random variable L cannot be 0 by definition? Is it because the string of days is infinite and there is always a possibility of at least having one losing day?
A string it must start with 1. If you start with zero , how do you know its gonna start because zero means its a success , so u have to get at least 1 failure before you start calculation of your string. Its a little bit intuitive. Just think about it for a while, you will get it
Hi Varun, you're right. The probability of having no losing days is 0. Having no losing days means that we have an infinite string of 1's. It was proven in this lecture that such an event has probability zero.
Yes, infinite string of 1's has 0 probability. But, also to pay attention to the definition of the r.v. L is "L is the distribution of the length of first string of losing days". So L is really meant to capture whenever you start having losing days and by that if you infinitely winning you wont even have L or we can say 0 is not in the distribution of L.
@@arpitb100 L is the length of the first losing sequence!!!!!!!!! Its value does not indicate anything about the sequence going out to infinity, only from the time you started observation to the first success.
I think hes just definiting it that way. Im not sure why it wouldnt be valid to consider a variable that goes to zero. In any case, its equivalent to the number of days youd have to wait since that first failure to get a success, so ig thats how L is a geometric RV.
he's not saying that the different processes are independent of each other, as they are indeed not (the top and bottom processes are dependent on the middle process (as there needs to be an arrival there for them to have a chance at an arrival) and on each other (as an arrival in the top means no arrival in the bottom and vice versa). However, each of the processes individually are Bernoulli process, the requirements for which include memorylessness, which he explains using that proof.
You miss the whole point. The Professor is not here to take you by hand through all the minute technical mathematical details for that you are supposed to refer to a textbook or technical reference. The take-home goal of the lecture is to cast a light on the most essential aspects of the theory in relation to real-world problems so that students may grow an intuitive understanding of the subject matter. Professor John Tsitsiklis delivers to his students what 1000 textbooks would never be able to it is an essential complement to reading the technical references.
Thank you MIT OCW, and thank you Prof Tsitsiklis. Your ability to explain so clearly is truly oustanding.
I followed through the whole series, and this is the BEST Statistics course I have ever taken. Thank you Prof Tsitsiklis for being so clear and concise. I had never imagined a starting statistics course enhancing my understanding so much and giving me so much confidence. This is magic!
this is how you read a lecture, thank you sir.
I've never come across a more eloquent, immaculate and lucid introduction to random processes before. It's a blessing to have the opportunity to listen to your lecture, Prof. Tsitsiklis. :)
Excellent clarity in preparing material and samples.
This is when the GOD guides you to find a treasure is to find such an amazing explanation, Thanks a ton.
He is excellent, indeed. His ability to explain simply is great
This lecturer is absolutely amazing.
Excellent way to introduce the concept of random process.
Outstanding explanation. Thank you so much.
Crystal clear explanation. Nice!
professor john tsitsiklis is a G.O.A.T
This is the best tutorial every on the subject
ha ha... Past Time of the Bernoulli's were to flip coins!! It made me laugh so loud!! Great humor sir!! :D :D
Eto haashar o kichu nei
One word. Thanks!
41:25 this is just negative binomial distribution.
Thanks for posting! @MIT
best conept
may all the goods and/or Governments bless u
Thanks a lot, great understanding.
While deriving distribution of time until kth arrival, it appears we have assumed that there will never be two arrivals in succession..! Does the argument change any bit if this assumption doesn't hold..? More so are we really correct to assume this at the first place?
wonderful professor
what's the sample space of geometric distribution?
38:38 I did not get what he meant. Y is a random variable indicating the time we have to wait. And for time, isn't it continuous? Then Y should also be a continuous RV. Why did he mention PMF rather than PDF?
R.V Y_k denote the number of seconds till the Kth arrival. If you think of time as number of seconds, then it is discrete.
Thanks thanks a lot
what happens in this slot, stays in this slot..
thxxx a lot a lot
Wow.
nice video
Technically the set of all infinite binary strings maps to the any countably infinite set such as the integers or the rational numbers, but not the real numbers.
Not sure about that. Even irrational numbers have binary representations (if you can make it in base 10, you can make it in base 2). I would think the set ({binary representations of all rationals} union {binary representations of all irrationals}) would biject pretty easily to the reals, and would be a subset of {all infinite binary strings}.
You're absolutely incorrect. The set of all infinite binary strings is not countable (in particular, it has the cardinality of the continuum).
infinite string is impossible to write down therefore uncountable
it is analagous to real numbers
unfortunately you're incorrect. this is a classic example in the first chapter of Rudin's analysis book
There are countable infinity such as integers and uncountable infinity such as real numbers. The Set chapter of mathematics for computer science discusses this difference.
30:30 why is professor saying that random variable L cannot be 0 by definition? Is it because the string of days is infinite and there is always a possibility of at least having one losing day?
A string it must start with 1. If you start with zero , how do you know its gonna start because zero means its a success , so u have to get at least 1 failure before you start calculation of your string. Its a little bit intuitive. Just think about it for a while, you will get it
Hi Varun, you're right.
The probability of having no losing days is 0. Having no losing days means that we have an infinite string of 1's. It was proven in this lecture that such an event has probability zero.
Yes, infinite string of 1's has 0 probability. But, also to pay attention to the definition of the r.v. L is "L is the distribution of the length of first string of losing days". So L is really meant to capture whenever you start having losing days and by that if you infinitely winning you wont even have L or we can say 0 is not in the distribution of L.
@@arpitb100 L is the length of the first losing sequence!!!!!!!!! Its value does not indicate anything about the sequence going out to infinity, only from the time you started observation to the first success.
I think hes just definiting it that way. Im not sure why it wouldnt be valid to consider a variable that goes to zero. In any case, its equivalent to the number of days youd have to wait since that first failure to get a success, so ig thats how L is a geometric RV.
Thanks a lol way much clearer
The verbal proof given at 46:22 seems to be fallible. Doesn't it argue for full mutual independence from pairwise independence?
he's not saying that the different processes are independent of each other, as they are indeed not (the top and bottom processes are dependent on the middle process (as there needs to be an arrival there for them to have a chance at an arrival) and on each other (as an arrival in the top means no arrival in the bottom and vice versa). However, each of the processes individually are Bernoulli process, the requirements for which include memorylessness, which he explains using that proof.
You miss the whole point. The Professor is not here to take you by hand through all the minute technical mathematical details for that you are supposed to refer to a textbook or technical reference. The take-home goal of the lecture is to cast a light on the most essential aspects of the theory in relation to real-world problems so that students may grow an intuitive understanding of the subject matter. Professor John Tsitsiklis delivers to his students what 1000 textbooks would never be able to it is an essential complement to reading the technical references.
"... it's because your sister knows..." Okay, that's creepy!
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