2. Introduction to Statistics (cont.) 

This is very cryptic, especially without the first part.

🎯 Key Takeaways for quick navigation: 00:00 📊 *The speaker discusses the central limit theorem and its application to estimating proportions using averages.* 02:19 🔄 *The central limit theorem implies that the distribution of square root of n times the average converges to a standard normal random variable.* 03:16 🔄 *The probability that a Gaussian random variable exceeds q alpha over 2 is equal to alpha.* 05:08 📉 *Confidence intervals are built by transforming the estimate in a way that results in a pivotal distribution, not dependent on unknown parameters.* 07:28 🧠 *The concept of pivotal distributions is introduced to create estimates that are asymptotically independent of unknown parameters.* 13:33 📉 *Hoeffding's inequality provides a useful tool for bounding the probability that the sample average deviates from its expectation for any n.* 16:47 📉 *Solving Hoeffding's inequality for a specific case shows a method for constructing confidence intervals without the need for large sample sizes.* 19:43 📈 *The Hoeffding inequality's worst-case scenario is discussed.* 20:12 🔄 *Combining intervals to determine the probability of a variable being outside a specified range.* 21:34 🤔 *Comparing the margin square root of log 2 over alpha divided by 2n to q alpha over 2/3n.* 22:55 🎲 *The role of assumptions in statistics, and the importance of balancing assumptions for confident statements.* 23:25 🔄 *Different types of convergence in statistics, emphasizing convergence in distribution as crucial.* 24:51 🔄 *Convergence in distribution explained through the concept of probability computations on random variables.* 26:17 🧐 *Stronger conditions needed for convergence in distribution to allow combining random variables effectively.* 27:42 🔄 *Almost sure convergence and convergence in probability explained, highlighting their differences.* 29:06 📉 *Convergence in Lp and its relation to the weakening of convergence conditions.* 31:27 🔄 *The importance of the characteristic function in proving convergence in distribution, especially in the central limit theorem.* 33:19 🔄 *Equivalence between convergence in almost sure, convergence in probability, and convergence in distribution.* 38:35 🔄 *The continuous mapping theorem: if Tn goes to T, then f(Tn) goes to f(T) for continuous functions f.* 39:32 📊 *Convergence in probability involves decreasing index values implying convergence, eventually leading to convergence in distribution.* 40:29 🔄 *Operations and limits differ between convergence almost surely and convergence in probability, allowing various manipulations for the former.* 41:55 📈 *Convergence in distribution criteria includes the convergence of characteristic functions or bounded continuous functions of the random variable.* 42:24 🔄 *Slutsky's theorem states that if one random variable converges in probability and another in distribution, certain operations are still valid.* 43:23 🚇 *Inter-arrival times in queuing theory, modeled by exponential distribution, are useful in systems with memoryless properties.* 45:16 📉 *Exponential distribution is often employed in modeling positive random variables, such as inter-arrival times.* 48:39 🔄 *The average of inter-arrival times in queuing theory, denoted by Tn bar, serves as an estimator for 1 over lambda, the rate parameter.* 50:32 🔄 *Strong and weak laws of large numbers support the convergence of Tn bar to 1 over lambda.* 52:49 📚 *The variance of the exponential distribution with parameter lambda is 1 over lambda squared, impacting the central limit theorem.* 56:53 📊 *Construction of a confidence interval for 1 over lambda involves manipulation of inequalities and dependency on Tn bar.* [01:00:11 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *📈 First-order Taylor expansion involves finding a theta bar between two values at which the expansion is performed, making them equal.* [01:00:38 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *📊 Multiplying by root n in Taylor expansion leads to root n Zn minus theta times g prime of theta bar.* [01:01:33 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *📉 Law of large numbers implies that theta bar converges to theta as Zn approaches theta.* [01:02:29 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *🍔 The "sandwich theorem" visualizes the convergence of theta bar to theta, considering the movement of Zn and theta.* [01:03:55 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *📜 Slutsky's theorem allows combining the convergence in distribution of Xn and convergence in probability of Yn if the limit of Yn is a constant.* [01:05:48 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *🔄 Slutsky's theorem facilitates combining sequences of random variables converging in distribution and probability under specific conditions.* [01:07:16 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *🔍 Delta method extends the central limit theorem to functions of averages, considering the derivative of the function.* [01:08:11 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *📊 Application of the Delta method to the function g(x) = 1/x results in a confidence interval for lambda.* [01:10:59 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *🔧 Replacing lambda by lambda hat in the confidence interval is justified by Slutsky's theorem, allowing for easier computation.* [01:12:22 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *🧮 Slutsky's theorem is crucial for replacing parameters with their estimates, simplifying computations in statistical applications.* [01:15:14 URL](ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-C_W1adH-NVE.html) *🎓 Understanding Slutsky's theorem enables the legitimate replacement of parameters with their estimates, maintaining convergence properties.* Made with HARPA AI

29:51 that’s why electron clouds are drawn as balloons with really solid surface areas. We can draw the surface of the solid that the electron is in 95% of the time (or whatever P is if it is not 95%) because of “convergence in probability”.

After reviewing probability courses and this one itself many times, I found this lecture to be a great one. Some vague knowledge in real analysis will help. But I still need to come back later after digging deep into other sound footings for foundation. Thank MIT and professor.

i'm a fan, professor! the breakdowns. the pacing. the reading of the room. noice.

The first lecture was not recorded. We are hoping to record it when the course is taught again in the Fall. In the meantime, please check out the course site here ocw.mit.edu/18-650F16 where you will find the slides that were used in all of the lectures.

Am I the only one who felt hard to follow the course? 6:31 means basically, using properties of variance, we have X bar as X_bar, the n observations of x are i.i.d. Bernoulli, Bernoulli's variance = E[x^2] - E[x]^2 = 1^2*p - 0^2*(1-p) - (1^2*p - 0^2*(1-p))^2 = p - p^2 = p(1-p). Having this variance, we can derive the variance of X_bar, which is var(X_bar) = var([X1+...+Xn]/n), know properties of variance, we take n out, as var(X_bar) = 1/n^2 * var(X1+...+Xn), since i.i.d., these xi all have the same variance, therefore we have n*var(xi) divided by n^2, so the variance(X_bar) = var(Xi)/n, so to standardize it (in CTL), we divide it by the square root of var(Xi)/n, which is what he wrote on the board. So, it all roots back to Prof. Tsitsiklis' Probability course.

This lecture is basically broken by the fact that there is a continuation mismatch to 1. - references to concepts and theories are made that were never discussed in the previous lecture.

Ahh dont you love when the teacher says "I am going to pick up this guy.." and the camera just zooms in to the point you CANT SEE WHAT THE "GUY" IS!!! ahhhhhhh so good bravo bravo.

Nothing is broken - go to the URL link get the lecture slides , get the HW assignments and everything you need to understand what is going on is there. Phillipe Rigollet is excellent and discusses with ease fairly complicated material. Excellent job. My two complaints are that the sound volume is a way too low - whoever recorded the first lectures did lousy job. Also the slides are not properly exported to pdf - there are some missing math symbols in some of the equations

was not expecting that scooter after watching part 1

The first video was recorded in Fall 2017. The rest of the lectures were recorded in Fall 2016, but video of Lecture 1 was not available. Only me notice this?

Why is sound volume so low?

You can get the slides if you go to the link in the description (and read the missing part in this video). Excelent lecture,it took me 3 days to understand all the concepts exposed. At 55:23 is not equal to alpha, is equal to (1- alpha) since that is the way he defined the q (you can see that in the slides).

When he talks about Hoeffding's inequality, shouldn't the tails be larger compared to a standard normal? I think smaller tails would result in a tighter upper bound (i.e. more confidence that the sample mean is close to mu), which wouldn't make sense when n is small.

The discontinuity from the first video adds quite some confusion.

how did it go from intro 1 to this? When did he tal about central klimit theorem etc? is he solving exercises? wtf

The sound level is too low, maybe re-upload with increased volume?

Oh no! Part 1 is missing! :(

I really liked the lecture. Thanks MIT!

Slide on 07:43 seems not correct. If we are targeting confidence interval of 1-alpha, shouldn't it be q_(alpha/2) and not q_alpha?

why did he ride a scooter during lecture?

A new intro stats series should be recorded either online (COVID baby!) or in person

I don't understand

better to watch the Khan Academy statistics playlist before MIT18.650

@7:00 ish.. If I am understanding correctly, limiting distribution is not always Gaussian. There might be cases when CLT isn't valid such as when n is small. Are there any other cases? And how were the bounds defined for kiss example? If someone can explain, I would appreciate it.

Thanks for the video professor

Pahtetic job of the sound recording ! Can't hear anything even with an external speaker

Of Anyone, Massachusetts "Institute for Technology" should be able to level the volume so we can hear it.. Breaking edge technology, but makes movies with 1990's PBS vibes.. WOW Thanks for the videos though!

I can barely hear the audio. Please remake this video next time.

Is this basic statics?? I am worried that i don't understand.... Can i have the stellar ink wheee the notes and assignment are loaded

Too bad this course is ruined by discontinuity... There seems to be a whole lecture missing, they should have added also the second lecture from 2017 at least. It's better to have some overlap than missing data.

I find these videos quite interesting, however the disorder at the time of writing that many teachers deal with bothers me. Unclear and messy handwriting. It would be nice if they could improve on those aspects.

maybe use a microphone next time

amazing lecture. Thank you very much.

At 0:22; you need a better duster

It's very kind of you

Can anyone explain what exactlt is alpha

in my years of college, i never seen the teacher describe complicated equation so deeply, most of them dont even know the details just put in the power point and explained and the student sleep queitly.

You'd think MIT would figure out the equation for chalkboard erasers. I guess that's more of a Harvard specialization.

any recommended books on understanding the mathematical basis for statistics as covered in this lecture?

No part uno?

Isn't CLT about the distribution of your estimated mean being normal (rather than calculating the mean from a very large sample)? i.e. calculate your mean 1000 times and those 1000 means will follow a normal distribution

@11:11:38 What does he meant lambda comes from here? Why does it come from the variance of the limit distribution? After this, where does the next expression come from, the one he divides lambda by, why does he do this?

Will the first part of this lecture be added as a separate video or will this video be replaced?

nice trike bro

Absolutely well done and definitely keep it up!!! 👍👍👍👍👍

Here I am trying to learn English, with a french accent teacher who teach statistics. What is the possibility I am be successful?

What's with the sound, guys??!?

Why is he using a bicycle?

Great lecture! I have enjoyed it; thanks.

Someone get that gent a chair and a overhead projector.

the quality is so low

Where did q and alpha come from?

He is probably the worst professor I ever saw to be completely honest.

I just remembered 18650 is a battery model 😂

Why can't we get the complete recordings of 2017 so it's continuous? :(

applied mathematics

1:01:08

Poor Recording.. can't hear anything

Aw come on, the sound is bad!!

anyone wrote the topics he had cover?

why is the professor on a scooter?

Thanks Mit

Test

why i can't watch it?

Introduction to brain damages.

WTF is this? i'm sorry, but this is not an intro class at all

Text book...........Please...!!!!!

I spent my years as a student. To teach something to someone, you firstly should motivate them by showing the point they will reach if they grasp taught. This is why firstly you should solve a lot of different real life example. In this course, students cant see the outlet of tunnel.

Ah college hasn’t changed! He started with an interesting example, then went into technical speak. He is now relying on equations he knows but the students don’t. He should be giving real common sense examples then using the equations. I am teaching AP stats and he just seems to jump into complicated equations. If this is a first year stats course I feel sorry for the students. It seems teaching at the university level hasn’t improved in almost 40 years.

Teaching is just a paycheck to this guy.

these statistics do not suit for psychology

How is this an intro?

Pretentious. MIT you can do better!

Totally useless......this shows that not everything that comes from MIT is useful

Come from 6.041 and this course seems too difficult for me..🥲