3. Probability Theory

MIT 18.S096 Topics in Mathematics with Applications in Finance, Fall 2013
View the complete course: ocw.mit.edu/18-S096F13
Instructor: Choongbum Lee
This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem.
*NOTE: Lecture 4 was not recorded.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@SeikoVanPaath]

Some notable Timestamps: 0:01:20 Random Variable (RV) 0:05:06 Probability & Expectation 0:09:01 Normal Distribution 0:25:32 Other Distributions 0:32:30 Moment Generating Function 0:48:00 Law of Large Numbers 1:04:00 Central Limit Theorem

[@ronakpatil1]

Thanks bruhhhhhhhhhhhhhhhh

[@viktorkhan8518]

Upvoted.

[@haashirashraf656]

It's amazing that this is for free, teaching done the right way whether your a high school kid looking for some deeper knowledge or even a college freshman trying to fully comprehend the basics or someone simply recapping basic probability theory, this video serves all purposes to some extent.

[@joshschwartz5622]

Thank you for the video. Just a note: you need to evaluate the moment generating function at t=0 after differentiating in order to get the k-th moment. It was implied, but not said. Thanks again!

[@woodypham6474]

This lecturer deliver a pain killer pill to students who used to be struggling to understand random walk and probability theory.

[@whatitmeans]

I think is more accurate to understand why Gaussian distribution is so universal because it is the maximum entropy distribution for a finite mean and variance, in simpler words, is the most dissordered possible scenario for a proccess with finite energy. It tells you that all information of the events is already lost, as example, like knowing the falling path of a ball in the Galton's board from the slot it have fallen. The lobe-like shape could be explained due concentration inequalities like Markov's.

[@abdelrahmangamal5875]

thanks for continuing uploading complete courses for free

[@abdelrahmangamal5875]

what're you talking about ?!

[@abdelrahmangamal5875]

Riemann Tensor are you mad ?! what's wrong with you ?!!!!!!!!

[@ObitoSigma]

abdalrahman mahdly He has a dream to get into MIT most of us. (You might already be a student for all I know!) He just expresses himself differently. ;)

[@riemanntensor8871]

Thank you! Look at my username, I love physics too!!!!!!

[@abdelrahmangamal5875]

oops .. freaking misunderstanding :D

[@alexpan5990]

two years ago , i could not understand at all because of my poor background, now i can follow due to my hard work on probability and statistics. Mr. Lee is awesome! Thanks for providing us with so good lectures!

[@WrathofMath]

Nice work! That's what it's all about, you work hard, you use the best resources you can find, and you get to enjoy the wonderful world of mathematics!

[@smuksm]

In a similar place as you Alex.. what lies next? Are you able to utilise the knowledge?

[@pathfinder2557]

Interesting. Did u get an IQ boost on the difference of knowledge? Proly not. U proly still have the same IQ as before but you are much more knowledgeable now.

[@DF-ed2jj]

There's something wrong at the beginning of the lecture. A random variable is a function from the sample space to R, that is X: omega --> R. Here's the guy said that are the pmf and pdf of a r.v. to take values from the sample space into R, which is uncorrect.

[@user-nx6uc7ot5t]

충범이 형님 수업 잘 들었습니다!

[@viniciuscorreadearaujofilh4946]

Thank you MIT.

[@albertrombone]

What is this guy experience with poker? We want to know more!

[@IVVIIVVII]

topppp. this lecture helps to understand probability logically by making theoretical ideas more sensible. def a battle all the way through. haha.

[@billdu1558]

Shouldn't the expression at 14:04 be (P[n] - P[n-1])/P[n-1] ?

[@subhanajiz3904]

yes, you'r right.

[@sidalitifoura620]

nah he's right.

[@lakshmikarle9371]

hi, can you share solutions to assignment problems please?

[@georgbraun7547]

There's an error at minute 10 - sigma^2 is the variance. sigma is the standard deviation.

[@94mathdude]

Couldn't agree more

[@dckdancing]

I also believe so..

[@epvtrinidad]

Exactly. Raised my eyebrow there

[@bikram_jha]

Yes

[@mohammadaljarrah7490]

In 2:38 it is not true that the p.m.f be a function from \Omega(sample space) to R+, the true is the p.m.f fX is a function from R to [0,1]. In fact the random variable X is a function from \Omega(sample space) to R, and the p.m.f fX associate to X is defined as fX(x) = P(s in \Omega | X(s)=x)

[@ehthsirig9402]

normalization makes it [0,1] buddy

[@yassinekened3138]

Thank you !

[@forKyrene]

2:05 Shouldn't it be Probability Density Function for continuous random variables? Or is probability density function the same as probability distribution function? As far as I know (correct if I'm right), probability mass function (discrete) and probability density function (continuous) are both probability distribution functions.

[@benediktwildoer8384]

Kyrene Says no, you are wrong.. The density Function is the last row at 3:30 .. The Density function [usual notation: F(x)] is the cumulated distribution Function [notation: f(x)]..

[@hmingthansangavangchhia4913]

Distribution function usually refers to the cumulative distribution function F(x). It's probability density function p.m.f for continuous and probability mass function p.m.f for discrete.

[@gouravban]

Thanks a lot.

[@haneulkim4902]

Law of large number seems so obvious since mean of r.v. is calculated via averaging all observations... So obviously if number of observation reaches # of obs that was used to calculate mean it will converge. Is my understanding correct? I'm doubting myself because it just seems too obvious...

[@ObitoSigma]

This is actually REALLY COOL and perfect for those just getting into Probability Theory. I love how he expresses himself with basic mathematics terminology for those not used to complex symbols. I'm currently 10 minutes in the video, but this is surprisingly *very interesting*. In fact, I might even take this course once I get accepted in MIT. It seems very feasible!

[@riemanntensor8871]

Obito Sigma A bit confident...ehh?

[@sujanbhandari783]

Riemann Tensor Dude You crazy or what?

[@15tefera]

did u get in then?

[@mtlotlomonasti3608]

omichael tmichael hahaha! Cracked me up

[@ObitoSigma]

@@15tefera Yes, I got in... believe it or not. Was a bit silly more me to say I might take this class since it's an 18.S class which means it's a special subject not normally taught. I'm a course 18C (mathematics with computer science) sophomore at MIT. Also, I can't believe that comment that 4 years ago.

[@aliyuismaila5511]

Thank you sir

[@math_person]

At 14:26 why is the variance of the normal distribution of P_n equal to square_root(n)?

[@shadhinreza6742]

Excellent

[@AdityaRaj-kt4ew]

To model the stock market, it is more reasonable to assert that the rate of change of the stock price has normal distribution (compared to the stock price itself having normal distribution). I don't understand why so?

[@konet1440]

When modeling stocks we are trying to predict how they will change. Stocks tend upwards with inflation of money/growth. If we assume that the price of a stock sits within a few values always oscillating in between, then we wouldn't be able to properly model the market. The main interest is the change in the stock. When googling the average daily changes in a bar graph a normal distribution may be observed.

[@mariushav]

If you took the price or a stock to have a normal distribution, you would also allow for negative stock prices. Research has found that a reasonable model for stock prices is the geometric brownian motion, defined via a stochastic differential equation. This is seen e.g in the Black&Scholes model

[@nikunjkedia7750]

where can i find the solutions of the assignmets?

[@digitalguard8672]

Took a few night courses. Was up all night with 3 problems. Thank you for helping me see the mistake I was making.

[@johanneswestman935]

If there's one thing that I learned in my engineering classes it is that theorems are fun and all but practically useless unless you're doing research. Monkey see, monkey do. Examples > all.

[@endgamme]

Just something I saw in the lecture notes on ocw link which states E[X^k] =(d^kM/dx^k)(0), shouldn't it be E[X^k] =(d^kM/dt^k)(0)?

[@shivanshmathur403]

Yes, it should be

[@OstapPetriv]

Is it for second cycle studies?

[@shabnamhaque2003]

What is epsilon at 59:44

[@janvisingh2587]

Where are these maths topics coming from😢😅 Any idea 💡? Where should I learn all these in hindi! 😅

[@Er.Sunil.Pedgaonkar]

Engineers are interested in applications of statistics & probability to their respective discipline,viz, Civil, Construction,Electrical,Mechanical, Electronics,Computer, Chemical, Aerospace, Nuclear,Marine, Metallurgical, Structural, Environmental Engineering

[@NgardSC]

I wish i had a teacher like him

[@user-ws8jm8uq4c]

you do here hh

[@benw4361]

When he says P(X

[@MiroslawHorbal]

I think it's a poor notation choice. P(X < x), eg, the probability that the random variable X is less than the fixed value x. For example, if X is distributed by a Log-Normal distribution, the expression: P(X < 3) would imply P( Y < log(3) ) for a Normal-Distributed random variable Y. Hope that helps :)

[@kushagraattrey2456]

Can someone share the next lecture the playlist doesn't has it

[@mitocw]

The lecture is not available. Since it was a guest speaker, it is probably due to IP. The topic was Matrix Primer taught by the Morgan Stanley Matrix Team. The lecture notes section has this written for lecture 4, "No lecture notes, but see The Morgan Stanley MatrixTM microsite for information about this topic", link: www.morganstanley.com/matrixinfo/. See the course for more info at: ocw.mit.edu/18-S096F13. Best wishes on your studies!

[@kushagraattrey2456]

@@mitocw thank you so much

[@user-ok4wr4zm5i]

The lecturer did not indicate that he used Chebyshev's inequality

[@Vamavid]

The only reason I understand this is I've done it before. I guess this means that MIT grads aren't smart because they went to MIT, they had to be smart to be allowed in!

[@suindude8149]

Its great but derivative always gives a fractional moment not positive integer....moment .....as log shaped exp also bell shaped.....but dispersion tells all......

[@ManishKumarmanimech]

Let A and B be two events such that the occurrence of A implies occurrence of B, But not vice-versa, then the correct relation between P(a) and P(b) is? a) P(A) < P(B) b) P(B) ≥ P(A) c) P(A) = P(B) d) P(A) ≥ P(B) Solution please

[@GustavTropoloYT]

b

[@nicoromero6423]

B. Because if the occurrence of A implies the occurrence of B but not vice versa, then we can say that A is a subset of B. In other words, B includes A, but there may be other outcomes that are included in B but not in A.

[@rivaldoaeynusy4738]

I'ts amzing that true

[@94mathdude]

mean of lognormal rv X is 0. Say Y~N(mu,sigma^2) and X=lnY. Then MGF of X is M_X(t)=E[e^(lnY t)]=E[Y^t] = integral over reals of some g(y,t) dy. Hence, as Y,t are independent, M'_X(t)= t*E[Y^(t-1)], so E[X]=M'_X(0)=0.

[@94mathdude]

oops my bad. Correction. M'_X(t)=E[Y^t lnY] so this doesnt give an easy solution to E[X]

[@94mathdude]

Sorry confused again. actually X=e^Y, so E[X]=M_Y(1)=e^{mu+1/2 sigma^2}

[@lemoi6462]

the mean of a lognormal rv X cannot be 0 since X always greater or bigger to 0.

[@ilikeandlovemathsandothers8880]

Congratulation

[@zarahussain5417]

I WISH you taught in the UK!

[@zunelmhrz3040]

I still don't understand lecture 2, 3, 4. How to apply this in finance????

[@andso7068]

Did you go through the entire course?

[@AntonioLopez8888]

Okay, so why here 1:11:30 Yn is exponential pdf? I personally know why, but I didn't hear it from him. This is due to Yn is equally expected at any point of time no matter what happened in the past. I don't remember exactly but either geometrical / poisson distribution, i.e. what is the probability if the event will happen in a certain number of trials.

[@summerQuanta]

He is writing the moment generating function (sometimes also called characteristic function as it completely characterize the distribution of a random variable). By definition this function has the exponential, he explains it at 0:32:30

[@tokitahmidinan2846]

I really dont understand what is a normal distribution just seeing the question of a problem

[@jacoboribilik3253]

In mathematical terms, the normal distribution or gaussian distribution is a probability density function that comes up a lot in a wealth of situations both in natural and social sciences. In order for you to understand what it is you first need to grasp the concept of probability density. In layman terms it is a function extremely useful for working out frequencies of events. If you have a bunch of people and you are interested in their height, the phenomenon can be well approximated by a ND in terms of frequency. The ND has a ton of important properties, by far the most crucial one is the Central Limit Theorem which mostly accounts for its presence in "random" processes.

[@gamer-lc8ip]

@@jacoboribilik3253 what defines random?

[@unpeacedralberteinsteinsze6395]

Most people are 5 ft 8 in Some are 5 ft 3 Some are 6 ft 2 There u go

[@unpeacedralberteinsteinsze6395]

@@jacoboribilik3253 random Some stock go overprice Some stock go underprice

[@kleinbogen]

Is there empirical evidence that % change in price data have a standard normal? In your video (at around 12 to 13 minutes), you mentioned that we want % change in price data to have a standard normal. However, what we want versus what is real can be very different. It may be convenient to use standard normal to come up with beautiful theories, do these theories stand the test of time?

[@benediktwildoer8384]

kleinbogen it is not... That is the whole Problem in accurate predictions and the reason why people can make money with financial instruments

[@benediktwildoer8384]

kleinbogen but: it is close enough why many people calculate with the stand Norm dev. .... But on the Other Hand this leads to crashes we saw in 2001, 2008, 2010...

[@benediktwildoer8384]

Models that calculate with other distributions Lead to much lower profits if no big crash or event happens... So for 99.9% of the time stand Norm dev. Is Quite OK, and the 0.01% really can fu*k over your model and in the end maybe the whole system :D so you cash in your profits and hope that no crash comes vor that you are out of the market a millisecond before it happens ;)

[@paul5324]

You defined the pmf and pdf using the sample space as the domain; I think that’s a bit misleading. You did mention quickly to just assume the sample is the real numbers, but that’s also misleading. The sample space may not contain numbers - for example if our random experiment is flipping a coin, then the sample space, say S, can be defined as containing the objects H and T for Heads and Tails, respectively. Thus the way you defined the functions f make no sense. It’s only when we define a random variable X, which is actually a function (borel measurable), such that we define X(c) = x for every c in S, x in Reals, i.e. X: S -> Reals. So in our example, we can define X(H) = 0 and X(T) = 1, and thus creating a space for X, say A where A contains the elements 0 and 1, which are numbers. This allows us to define a pmf correctly now: f_X : A -> Reals. If I got this wrong, my apologies, but this is how I remember it.

[@juanguang5633]

4:52 fx(y)=1 for all y? is that a mistake?

[@whatitmeans]

the uniform distribution from [0, a] with a>0 gives you a f_x(x)=1/a such its integral in [0, a] gives you the value 1. Just happens that choosing a=1 gives you f_x(x) = 1 (its logic, but kind of counterintuitive at first glance).

[@litoboy5]

COOL

[@TheFreshErniOfBremen]

which topic has lecture 4 been?

[@mitocw]

The topic for lecture 4 was "Matrix Primer". See the course on MIT OpenCourseWare for more information at ocw.mit.edu/18-S096F13.

[@TheFreshErniOfBremen]

ok thank you very much

[@AntonioLopez8888]

@@mitocw no such lecture there

[@davidsoto4394]

They should use a dry-erase board because writing on the chalkboard makes it difficult to read.

[@stupidpoor5004]

i love this graffiti artist gg Mr lee

[@jftsang]

What was in lecture 4?

[@mitocw]

Lecture 4 is not available. The Lecture 4 topic was "Matrix Primer" by Morgan Stanley Matrix Team. See ocw.mit.edu/18-S096F13 for more info. Best wishes on your studies!

[@matildeguadalupecerdaruiz1340]

Which textbook do u use?

[@mitocw]

There doesn't appear to be a textbook for this course. We see case studies and lecture notes. See the course on MIT OpenCourseWare for info at: ocw.mit.edu/18-S096F13. Best wishes on your studies!

[@zl7460]

so trivial

[@jiteshbohra6164]

the class is empty cause of the last lecture!

[@WallaceRoseVincent]

Anyone interested in working through the course together?

[@guhanpurushothaman9313]

I am. My instagram is instagram.com/guhanpurushothaman/

[@ibrokhimqosimkhodjaev6326]

me. But, I think i am late)

[@WallaceRoseVincent]

@@ibrokhimqosimkhodjaev6326 No you are not late. I'm just not sure if it's possible. What's your goal?

[@WallaceRoseVincent]

@@enisten Yes. Can you watch this comment location so we maintain communications? What is your name? What is your location?

[@WallaceRoseVincent]

@@ibrokhimqosimkhodjaev6326 it isn't that you are late, it's that it is difficult to connect via comments on KZbin. ☹️

[@Spectre.007]

no Lecture 4?

[@mitocw]

*NOTE: Lecture 4 was not recorded.

[@Spectre.007]

@@mitocw May I know the Lecture 4 topic title? Thank You.

[@mitocw]

Lecture 4's topic was Matrix Primer with the lecturers being the Morgan Stanley Matrix Team. See the course on MIT OpenCourseWare for more info at: ocw.mit.edu/18-S096F13. Best wishes on your studies!

[@benediktwildoer8384]

I know that it is a little fast in General, but am i the only one who is amazed, that he can put a whole year of high-school math-classes Into a 90min session? And you can actually follow what he is talking about??

[@mynewnameisbeautiful___4717]

I didn't understand anything

[@mathandmath.]

Nghiên cứu hàm số❤❤❤❤

[@PapaKakaes]

4:51 "...this is some basic stuff"

[@rysknet]

From this comment I was expecting him to dive into something crazy. All he was going was letting you know what notation he was using to represent each function. It’s actually helpful because if he did jump right into it without explaining the notation it might get confusing.

[@peterd5843]

48:00

[@devesh3648]

Where is lecture 4 bro????????????????????????????????????//

[@mitocw]

Lecture 4 is not available. The topic was "Matrix Primer" done by the Morgan Stanley Matrix Team. It's possible they didn't sign the IP forms, or were not happy with the video? It could have also been because of technical issues (no audio, crew missed the lecture, video file got lost, etc.)? There is no note on the course by the course authors.

[@devesh3648]

@@mitocw Genuinely appreciate your clarification. Thank you :)

[@user-ok4wr4zm5i]

what is this lecture consisting of definitions and theorems?

[@mohammedouallal2]

Teaching is golden skill that is not given to anyone. This doctor, is definitely brilliant in what he does, except Teaching

[@Killakane23]

Are there any solutions to the problem sets?

[@mitocw]

Sorry this course does not have solutions for the problem sets. See the course on MIT OpenCourseWare for more details at ocw.mit.edu/18-S096F13.

[@mathandmath.]

Nhóm toán❤❤❤❤❤❤❤❤❤❤❤

[@user-oe2un9yh1m]

The only thing what I don't like in this video is the dirty board eraser.

[@AshishPatel-yq4xc]

Very difficult to follow and I've done some probability stuff before but the way its explained here, the whole thing is a mess.

[@paulkane1535]

No it ain’t, he just does proofs by definition after an example. Get your math right. It’s you not him.

[@peterd5843]

Algebruh

[@bobby4360]

Those gainz though

[@chebonrunner3422]

Was this a timed trial? You could go faster if you just pretend you are the only one listening. (Trying to be funny about it, but your lecture is good, but your speed and penmanship render the lecture nearly noise, UNLESS you already know the topic.)

[@CC-qt5kd]

Want to help him erasing the blackboard lol

[@thankor]

I was following right up until 0:36 then I was lost.

[@GoodaJayz]

😂😂😂

[@_Sam_-zh7sw]

I am 27 min into the video. i have learnt differentiation and integration of multivariate functions and this lecture still sounds latin to me....On the course page it says that knowledge of linear algebra,calculus and statistics is not required.....

[@denizaydn7716]

poor guy, i wish he had had sth to clean that dusty board

[@amirmn7]

so many mistakes, can't follow :(

[@mohammedouallal2]

Teaching is not given to anyone!

[@alan713812]

how to be as smart as him

[@pathfinder2557]

yo proly not gonna be as smart as him. the guy was a summa cum laude as an undergrad, holds phd from ucla and most of all he's an ASIAN and yo know the reputation of asians when it comes to maths

[@masteroogway8601]

pusing anjeeenngg

[@jordym9999]

Some guys are just not meant to teach. Compare this to Prof. Andrew Lo (his course Financial Markets I available on this channel) to see what I mean. Thankful for free courses anyway!

[@MasayoMusic]

Is it math heavy? Do you have a link to the playlist?

[@user-ug8cn3ze2o]

Ahhaha

[@user-ok4wr4zm5i]

confusing explanation

[@benjaminlittle9905]

At UCI, we had WAYY BETTER probability courses. Everyone looks to KZbin to find MIT’s version well in this case I’d tell the MIT version to jump in the lake!

[@steve6012]

This is not a probability course

[@abhishekvanenooru2869]

HARD TO UNDERSTAND YOUR LECTURES

[@muradmath]

Lol, you don’t have to take them

[@surajshukla1477]

He must be the worst professor at MIT

[@R3b0rNz]

u sounded like a loser

[@saiyanuigod9568]

@Digital Nomad yea for graduate student it is normal as the material is already teach on undergraduate, but hey no hurt to re learn all the basics too. This korean clearly either want to show off or the students' are prick that want to speed up the teaching, as that class is already decided to be put on OCW. wtf men

[@lekshmipriyap2932]

Not good... Please prepare well before taking classes,

[@robertwanko219]

choongbum? seriously?

[@tejprakash3561]

This course assumes too much. Uses terms without explanation. Writing on the board is no explanation. not very useful.

[@tuhinmukherjee8141]

The teaching is very messy tbh

[@test8352]

this guy is smart but over complicates simple things with unnecessary mathematical jargon. He could cut the notation by 95% and still arrive at the same conclusion. He is notating to show off how smart he is.

[@benediktwildoer8384]

test this is not elementary School.. This is University.. It is not flower-power schooling, but science.. And people who pay 100.000$ a year in tuition are expected to be able to either follow him or be so smart/interested in the field to work it out for themselves after Class..

[@PyMoondra]

He’s a math researcher. When you get to his level you understand the importance of every little detail. It becomes a natural awareness.

[@test8352]

@@benediktwildoer8384 ah your one of those elitist assholes who prob goes to a rinky dinky no name school. Got it.

[@someone20ify]

he is not a very good teacher. no offence

[@daniel24ful]

Thank you!

[@BubuRulez]

Thank you!