Poisson process 1 | Probability and Statistics | Khan Academy
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- Опубликовано: 9 сен 2024
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Introduction to Poisson Processes and the Poisson Distribution.
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First time ever i couldn't understand sumthin in khan academy
Completely agree; so lost
i blame it on the video quality
Same here. :/
same
Concur.. that is why I came here to commiserate with fellow khanists
I had to watch like 4 times to finally get it, but finally I got it. I think if you understand the difference between lambda, N and K you should be able to follow it.
Ignoring Lambda and only concentrating on K and N for now; K is NOT the number of cars that passes, it's the number of MINUTES that a car has PASSED.
Every minute counts has 1 observation. So 60 minutes = 60 observations.
If you want to calculate P(X=60) you want to calculate the probability that of 60 observations there were 60 passes.
There is a pass when AT LEAST one car passed by in ONE minute. Every minute is ONE OBSERVATION. So 60 passes out of 60 observations means you want to know the probability that at least one car passes every observation of the total number of observations. One observation = one minute, so you'd want to calculate the probability that AT LEAST ONE CAR PASSES every minute of the 60 minutes.
Now heres the catch; there is a PASS/SUCCES when AT LEAST ONE CAR passes in a minute. What about the fact that multiple cars could pass in a minute? Shouldn't that be taken into account? That is where Sal continuous explaining; alright, we could make this more granular by not looking at a pass per minute, but per second. That way multiple cars are taken account for. But what if multiple cars pass per second? Well that we could make it even more granular by checking a pass per millisecond. But what if?.. etc etc, the most accurate approach would be reaching the possibility of successes over a time frame that could get so close to 0.0000000000000000000000... seconds that you could assume infinity. Now how do we take away that time constraing influencing the outcome of our probability calculation? That is explained here by Sal, by introducign the constant e.
Thank you for the comment, it made me understand Sal better.
Thank you for your explanaition
IngamerX thanks a lot mate! That really helped haha
Thanks for that who ever your mystery tutor.
Thanks, brother!
Excellent video and step by step illustration of using the binomial distribution to derive the poisson distribution. This is so much clearer than in my undergraduate stats class!
Ok here is the explanation for everyone and i know it’s late. You could imagine that the binomial distribution is for either happening or not happening and for n trials. Now we are doing the same but with a time period of one second. The probability of the « happening » in one second is lumbda (i’m referring to it as k) and you might say: yeah i can also find the probability for 1 hour or even for 1 millisecond or even for 1 moment, it would just depend on how many time intervals there are in a second or the opposite, then we could say k= np as p is the probability of the « happening in each of these time intervals contained in the second we are studying, now imagine that we consider that for every moment, there would be a probability of the event happening but which would be very small. So understanding that we are dividing the probability in a second or an hour or a minute by a very big number of very small time intervals is very crucial. The poisson process is basically a binomial process for a gigantic amount of very small time intervals where the probability is very small. So the limit of the binomial process formula when n is +oo and p goes to 0 is the formula for the binomial. Reason why we do this: when we are trying for a second, we are actually trying for every moment of it, with a small probability for every moment. Hope it helps, sorry i’m first year college and it’s the first time i see the video and comments ❤
Of course giving one probability for a number of trials means we are giving the mean; which is why, lumbda is a mean value. It’s obvious since even velocities are mean velocities for time intervals
For those who are struggling to analyse the K implies....
Assume there are 60 interavals min 1 , min 2 , min 3.......min 60. This is analogues to 10 basketball shots... Shot 1 shot 2 shot 3.... Shot 10 in the previous example.
Now what does P(X=1) mean in BB example? Probability of ( # of one success shot out of 10 shots)
Similarly.... Here P(X=1 ) means probability of ( # of 1 success min out of 60)....if a min is success min if a car is passes in that min.
Generalising: K implies Number of success minutes out of 60 minutes
For those who are struggling to analyse what the K implies....
Assume there are 60 interavals min 1 , min 2 , min 3.......min 60. This is analogues to 10 basketball shots... Shot 1 shot 2 shot 3.... Shot 10 in the previous example.
Now what does P(X=1) mean in BB example? Probability of ( # of one success shot out of 10 shots)
Similarly.... Here P(X=1 ) means probability of ( # of 1 success min out of 60)....if a min is success min if a car is passes in that min.
Generalising: K implies Number of success minutes out of 60 minutes
wow It is amazing to learn that just car passing events which look irrelevant to binomial can be expressed by binomial. Thank you.
Once again! I found that little piece of information I was missing, how beautiful he connected the λ parameter with the binomial distribution. Excellent job!
I had to watch maybe 15 videos about functions, limits, compound interests, and e to understand what is going on in this video, but it feels like an adventure, and finally, when I beat up this video it feels like I just beat up some dark souls boss, feels good.
Ok here is the explanation and i know it’s 7 months later. You could imagine that the binomial distribution is for either happening or not happening and for n trials. Now we are doing the same but with a time period of one second. The probability of the « happening » in one second is lumbda (i’m referring to it as k) and you might say: yeah i can also find the probability for 1 hour or even for 1 millisecond or even for 1 moment, it would just depend on how many time intervals there are in a second or the opposite, then we could say k= np as p is the probability of the « happening in each of these time intervals contained in the second we are studying, now imagine that we consider that for every moment, there would be a probability of the event happening but which would be very small. So understanding that we are dividing the probability in a second or an hour or a minute by a very big number of very small time intervals is very crucial. The poisson process is basically a binomial process for a gigantic amount of very small time intervals where the probability is very small. So the limit of the binomial process formula when n is +oo and p goes to 0 is the formula for the binomial. Reason why we do this: when we are trying for a second, we are actually trying for every moment of it, with a small probability for every moment. Hope it helps ❤
rewriting the binomial distribution in terms of successful seconds is interesting and seems to me the coolest idea. Thank you for sharing.
I couldn't able to get this video
For the first time I couldn't able to understand video from khan academy.
This world would not hv been the same without Khan's Academy
Sal, you are truely a philanthropist 💯💯💯
Crystal clear! Thank you for this amazing video
Hey Sal, I want to thank you for doing these videos and also present a request. Would you please make some videos of other forms of distributions like Gamma, Beta, Pareto, Etc.?
Hope you're in a better place now
oh god, amazing approach to explain poisson!!! Thank you so much!
probability for something to happen twice in a row :
p^2
probability for something to happen k times in a row:
p^k
probability for something to NOT happen k times in a row:
(1-p)^k
where p is 0
He is better than my double phd professor.
Great video thanks!!
sometimes u explain things carefully... sometimes you just skim through
Thanks Arther. I figured it out. Passed my class with a respectable B+ for an old gal like me. Made me realize I should study more as I have children with homework and math/sciences are really cool.
khan is the best teacher in the world 🙏🏻👏🏻👏🏻👏🏻👏🏻👏🏻👏🏻👏🏻👏🏻👏🏻👏🏻👏🏻
I was able to under the Poisson process from you much easier than my stats prof, thanks!
It gives good intuition about how poisson distribution has its form.
beautiful how you show the evolution of the idea
Please make a video on hypergeometric distribution and negative binomial distribution.
Explaining by story telling is the best. Thank you
dadada, then you could be unstoppable!!
you are the hero!
Stats be the death of me. Its all seems so counter intuitive to me but I really hope to grasp it.
Good video - my question is answered.
How Poisson distribution is derived ?
Thought it was just me feeling lost :/
In fact, the explanation is good. The problematic is statistics, it is quite confusing by nature; most of the time, not to say all, statistics goes against our senses.
+Jorge Pires these results are very basic with an understanding of the binomial distribution (literally heads or tails and the probabilities) and limits in calculus
Tyler Hushour I am not sure you understood my comment, it was pretty generic, likely placed in the wrong item. Nonetheless, thanks for your feedback. It seems pretty sensible for the video context. Statistics is a game of taken-as-granted, if you starts to make sense, you lose it. For me something is easy and sensible when you can remember it after years. I have been working with statistics for years and I always need to review some concepts, different from physics and calculus.
I used to share that point of view that statistics is always counter intuitive because it's really different from the way we think as human beings. But don't let that fool you, prob is still easy on paper, you just can't lose track of definitions and the basic axioms or other properties
Tyler Hushour Okay, you may be right, likely it is my fault. Best regards, Jorge Pires,
Tyler Hushour Let me correct one point. By no means I think statistics is counter-intuitive, probably a miscommunication matter. I believe people see it as counter-intuitive due to the fact we have been addicted to a picture of reality which is not true. For instance, it is quite hard to learn quantum mechanics, mainly because most of the ideas of Newton will not be valid, such as momentum, for instance it is easier to trace the absence of an electron than its presence, it is easier to assign probability to a particle than its true position. All this fly against the fact of reality we have. Nonetheless, I still believe that statistics is counter-intuitive on a daily basis, not because it is wrong, but because it is not "taken seriously". It is almost impossible to explain someone the meaning of confidence interval. Of course, if you work with that everyday, it becomes easy to grasp, and it becomes your life. I do not believe I have been fooled by my view, neither I believe you have been, they are just different perspectives; I am an engineering, I had four university boring courses on that, and one master course on that. The fact that I do not believe in god does not mean I cannot have faith, they are different things. Feel free to give me a response. (if you have already received my feedback, sorry, I tried to remove a link for you to see it)
is the poisson distribution on the irish leaving cert course????????
I am trying to find out what Poisson Regression is/how it is used, i am doing this for an assignment which is based on Research Methods, i am not a math student, nor am i a pyschology student, i have never used and or heard of this methods, i am struggling to write about it because all of the information i am finding is too in depth for my knowledge of the method.
I just wanted to leave this comment, because i watched the entire video and could DEFINATELY see this being useful fo anyone.
Thanks
It's hard to understand each variable what is n and k
why 60 minute/hour is become the n? why 60 minute/hour means 60 trials? what is k over the n?
Trial is seing or not a car passing within a given interval, being minute here. So, 60 trials max. within an hour.
k, number of successful trials, ie. a car actually being seen.
no mention of the markovian assumption but still compelling stuff!
I'm looking for the same thing, Rahul, did you find any good videos about Weibull?
the first assumption is wrong. There is a time period you investigate, and assume independence in that period only. The next hour etc might have a totally different lambda.
Excellent
thnaks
Dude, until yesterday the Poisson formula was complete black magic f*ckery to me. Thank you very much for shedding light over the big question mark I have had over my head for an entire week!
Great video!
We are having problems with chebyshev's theorem in probability. Can you please do something about that. I would be very thankful for it
Before watching your videos, I though the NewBoston was the best educational channel on RUclips, but apparently I'm wrong.
D to E to T to O to HOLD UP....X 07:35
I read that the events can take place anywhere in space and time, so it can be blemishes every 100 metres of carpet or cars passing every 100 hours.
It then went on to say that the events can't take place at the same time? Well, I'm sure that the probability of getting 3 blemishes every 100 meters of carpet would be the same, whether me and a friend looked at different 100 metre sections at the same time, or if I looked at each section one after the other?
Why we can take it as a binomial distribution in the very beginning.
awesome video!!
love videos from you!
Hello Sir, you have mentioned that "n" is no. of trials and have equated to 60 min/hr, but @ 4:30, u say choose K=3(no. of cars) from "n". It seems ambiguous. Kindly explain.
Hey the title says Poisson Process, but this video is about Poisson variable only, they just don't know what they are talking about
In the earlier part of the video, think of each second as a "trial" and a car passing through during that second as a "success". Then, you can treat it like any other binomial distribution
What is the specific video on compound interest he's referring to? Anyone know?
Wow Khan Academy FTW!
will be great !
good question :)
thanks for this! Eaasyy to understand! When looking it up on other sources I was like: WTH?Indepedent variables in a fixed time interval or spatial entity whatever... now things come together. like how one can imagine what poisson is and from where the densities originate... thank you
6:31 *POISSON*
Why is "1 / n = a / x"?
is lambda - number cars PER hour, or more like IN an hour? I mean "PER" sounds more like for describing frequency, isn't it?
It seems that you're deriving the Poisson distribution, not the Poisson process. For example, the Gaussian PDF is an important part of the Gaussian process, but the latter is MUCH more complicated.
This is way more confusing than my actual stats class. :(
you probably dont care at all but does anyone know of a method to log back into an instagram account?
I somehow lost the login password. I would appreciate any assistance you can offer me.
@@iancolin1978 why?
@@iancolin1978 just google lmao - why are you commenting this on an 8-year-old comment?
8:00 You replace x with n*a, then you say as x approaches infinity, n approaches infinity. That's true if a is positive. But what if a is negative, as in your next video?
This is exactly the most confusing point to me…
On the other hand, using the fact that lim_{x -> infinity} ( 1 - 1 / x)^x = e^-1, we can solve this part easily.
i dont understand where the lim (....) comes from??? Anyone can explain this for me please ???
10 years later
great videos
nice I like how you are branching deeper into probability theory, could stochastic be on the horizon?
This is so hard
I use saylor.org and augment jy kearning with Khan.
SLOW down please. Always these math teachers that are A: worst at teaching, B: unable to contain their own excitement. I'm not excited. I NEED to learn this, so please, curb your enthousiasm mister teacher.
I don´t understand where did the a,b, and e come from?!!
06:47 'Flipping coins, that's where everything is coming from'. Fundamental truth of the universe finally uncovered! Thanks Sal!
the only problem im having is why the probably a car passes per min is lambda/60
Edit: oh wait it was bcuz of the lambda = np written above
Did you assume that there is only one car passing in each minute? what if, say, lambda is 600, which is way over 60, then we want to calculate the probability of exactly 100 cars passing through? then 60 Choose 100 is not defined.
what a fancy saying of poisson at 6:31 ! really liked it
why is the probability of success = lambda/60 ? isnt lambda just the expected value?
pOISSON distribution
lmao "THE POISSOÑÑ"
Does anyone know how he makes these videos?
I feel this is a wrong example .... n is number of experiments/trails and it is the random variable ... The probability should be calculated when P(n=k)... Is that right?
Ya but it also applicable to traffic too
2:24 Half-Life 3 confirmed!!!!!!
And I though 8th grade was hard 💀💀high school has killed me this year
Google Khan Academy and scroll down for other subjects that Khan speaks on personally:
Physics, Biology, Chemistry, Math (algebra, Calc, Trig, Stats, etc, etc), Art History, Computer Science, Economics, Finance, History, SAT Preparation, etc, etc, etc.
If your in school take a look and see if Khan has talked about the subject. He's financed by the Gates foundation and Google, etc.
His vision: Eventually have all university level education online and below for free
The what distribution?!!
why are u doing 60 pick k. It's like saying you're picking 3 cars from 60 minutes. Comparing two thing that don't match. and what of the case you want to know if more than 60 cars pass in that 60 minute interval? Do you then do 60 pick 100? which is a probability that always equals 0. And 100 cars passing is definitely possible in a real life situation.
Edit: Nvm, the rest of the video explained it.
05:08 .. i did not understand that part ... why is he breaking it into seconds and milliseconds ? How does lamda / 60 does not work if there are 2 cars passing per minute
In an hour, there is 60 minutes, which we consider as 60 trials.
What the actual f*ck...
4:07 is where it starts to fuck with your head
7:14 is where you would say "fuck. this. shit" imma kms
4:07 is only the binomial distribution...
why is it the expected value and not the mean?
+Erin McCullough Expected value is the "expected" mean after an infinite number of iteration. Mean is calculated when the number of sample size is "known." However in an infinite population the number of the size is unknown. So after an infinite number of iteration, you can "expect" the next value to be somewhere near the mean of the infinite population, so you "expect" the value. The word mean implies that you know the sample size so you can't really use it for a sample size(population) with infinite number. Expected value refers to the mean of an infinite size cause you would 'expect' that value. This is my wild guess "why" they(statisticians) call it an expected mean.
I hate statistics especially stochastic and random processes...Full of Confusion.
Poisson is easier to calculate and less accurate.
What
The
Hell
I dont understand :/
KHAN SIR I REALLY ENJOYED UA PREVIOUS LECTURES BUT THIS ONE I DID NOT
I tried to substitute n to a really big number into the equation, but i came to an answer of 1 instead of e.????????????
Your value of n could have been too small for your calculator to deal with. It would then have 1 + n as being 1, which taken to the nth power is 1.
You have to be sure to include the limit. Then it will work.
What is the first name before Khan?
His full real name
Salman Khan
Cas désesperé mon vieux!
This is not the way to explain Poisson distribution :). But rest all videos are nice for a quick revision.
What
became greek to me after the analogy of the cars.... completely lost
I did not understand why two cars screw up with the stuff
I believe that is because you cannot approximate it with the binomial distribution. The binomial distribution only allows for two outcomes, either 1 car passes by in that interval or it doesn't. 2 cars passing by is outside the sample space and so you need to shrink the time interval until it is impossible for 2 cars or more to pass by.
That's quite the way it is. The Binomial distribution only allows for Bernoulli tests, which yield either a success or a failure. Once you can have more than one success/failure in a single test (i.e., the test being about the second, the millisecond, the nanosecond, or whatever), Bernoulli tests fail to represent reality to an acceptable accuracy. Hence, the need for Poisson to come and save our days ;D
don't mean to be rude or something but this is way more complicated than patrickjmt's version.
this version is asking me to go back and start with binomial distributions again if i want to understand.