5. Discrete Random Variables I 

"A random variable is not random. It's not a variable. It's just a function from the sample space to the real numbers." Good stuff!

I thought that i'm not smart to understand probability, but turned out that it depends on who is teaching it. this prof. is great.

28:00 The expectation of a random variable is just the center of gravity. This is a killing point of this class, literally explaining what expected value is and how it naturally be used. I have never been this jealous of students at MIT until I see someone explaining the logic of expectation in a sentence like this.

0:54 Outline 2:36 Random Variable 24:53 Expectation 43:00 Variance

Your teaching is amazing. I am a teacher myself and I know that teaching is not easy but it seems very easy to you. Thank you for these videos. I am very grateful for them :)

This man is extremely clear, excellent lecture!

Excellent explanation about Random variables. Random variables are just dunctions or simply a routine which give us some numerical value.

amazing teaching methodology...salute sir!:)

the best maths teacher ive come across!

So glad I came across this!! Thank you so much MIT!

(28:13) Regarding the analogy with the center of gravity of a light rod with weights at discrete positions, if you'd like to visualize the balancing, the position marks should be further apart, 2 a unit to the right from 1, and 4 further two units to the right.

Really sir want to thank for your great job.

Great lecture, really concise.

That is why MIT is called the "Mecca of Engineering". Brilliant !!!!!!!!!

This lesson is so great! Thank you so much!!!

This guy is really good. Wish my professor could articulate the material like this!

Thank you very much .You are great teachers

Wonderful Explanation!

OMG! Amazing teaching!

This is very good.

Great lecture

This video is a masterpiece!!!

I love the way this guy says "way" and "when"

good informative lecture .

Would be nice to make an insight on the functions over random variables. If i have Y=e^X for example

Thanks,very clear

Python scipy + matplotlib: import matplotlib.pyplot as plt import scipy.misc import math def binomial(n,k,p): nk = scipy.misc.comb(n,k) return nk * math.pow(p,k) * math.pow((1-p),n-k) def getProbabilityList(n, p, calcFun): vals = [] for i in range(n+1): vals.append(calcFun(n,i,p)) return vals def PlotProbabiltyRange(n,p): x = range(n+1) y = probRange(n,p, prob) plot(x,y) def main(): n = 100 p = 0.5 for i in range(1,n+1,1): y = getProbabilityList(i,p, binomial) x = range(len(y)) plt.plot(x, y) plt.show()

whoooo!!, just lost in the wild. I have to watch it again.

This guy is great.

Good lecturer.

i ask our lourd to preseve you and your whole familly .and thanks MIT open courseware for all

beautiful

Thank a ton Sir.

Omg Professor, you are awesome,where is the abstract concept when you explain them in reference to reality!

Great video! Just one question. At the very end when talking about Variances. I know he said it is two lines of algebra, but how did he get from the sum of [(distance from the mean) * p(X)] to E[X^2] - [E(X)]^2 ??

THANK YOU SIR!

Does anyone else have a pet peeve with the mathematical notation used for numerical value generated by the random variable as g(x) at 41:00? The professor says at 9:10 that "x" (in lower case) refers to the numerical value generated by the random variable. Why does he use g(x) for it later? Loosely speaking, he could have referred to it as g(sample space) as well, and although it isn't strictly a function of the sample space, it at least honours the the initial conception of "x" as the value generated by the random variable (which can be thought of as a function of the sequence(s) of events that satisfy the constraint g(sample space)=x).

Randomness is relative, it does not have a absolute meaning. It is wrong to say that something is random or that something is not random. You could say that something is more random or less random depending on the criteria that is being used. You could even eventually consider a scale do measure "randomness" depending on the outcome distribution type of a variable. The true meaning of "randomness" would be, for instance, a discrete variable whose possible outcomes have EXACTLY the same probability of happening. The more random a discrete variable is the more likely it's outcome distribution follows a certain type (independently of the specific values that it will assume). A uncertain outcome of a discrete variable may reveal or prove itself as being more or less random considering the type of outcome distribution that it will take. All the exercises that are done resolving binominal, polinominal, multinominal, call them what you want distributions are assuming that all possible values of a discrete variable have exactly the same probability/chance of happening. In other words they' re assuming that the variable is actually random to it's fullest extent. The truth is that the correct answer to these problems is only theoretic because in application it would be impossible in most cases to replicate randomness. This would mean that the actual practical, real life answer to all these exercises would have to consider a more likely convergent distribution type in comparison to the more likely theoretic random distribution type. A random variable is indeed a variable but should not be initially considered random until proven, through a certain number of trials (the more the better), to have a more likely theoretical random outcome distribution type . A random variable could be associated with a function (with no input values and with a given range of possible outcome values that follow a certain distribution type). The possible values that a discrete variable could take doesn't necessarily have to be numbers. They could be anything including shapes, colors, objects, symbols. In other words they could represent anything. This would be the big difference compared to continuous variables.Take the example of dices that, as everybody knows, assume the following outcomes "1,2,3,4,5,6". You could consider the numbers as being symbols and not numbers. In this particular case the outcome values "1,2,3,4,5,6" could be looked at as being symbols without any arithmetic value. In this case any correlation between the sequence of these symbols and their distribution should be seen as a coincidence. Meaning that there is no sense to calculate the mean or the variance in this particular case, because there is no arithmetic correlation value between symbols or things independently of a more or less random outcome.

9:24 so glad for this clarification, was confusing me while reading a different textbook that didn't quite cover that detail. P(X = x) huh?

This professor is just amazing

I do not understand the fact that why the summation of expected values gives mean?? Could u please help Thanks in advance.

Thank u sir. Hope u read this !!

Was there an inconsistency in what he did at 34:33 and 34:37, he pointed to different lines? It seemed intentional, can anyone explain why he multiplied by a number on the "y=g(X)-axis" and then a number on the "x-axis" when everything maps to the same number on the y-axis?

link to lecture notes ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/video-lectures/lecture-5-discrete-variables-probability-expectations/MIT6_041F10_L05.pdf

This is my MATLAB code for his example at 23:35 min, ---------------------------------------------------------------------------- clear all clc outcome = 2; % outcome # of a coin (Head & Tail) n = 4; % tossing # k = 0:n; % # of getting Head or Tail ( zero means no Head or no Tail) p = 1/outcome; % the probability of a single outcome Px = zeros(1, size(k,2)); % store the results in a vector for i = 1:size(k,2) Px(i) = nchoosek(n,k(i))* p^k(i) * (1-p)^(n-k(i)); end plot(k, Px) xlim([ min(k), max(k)]) ylim([min(Px), max(Px)]) ----------------------------------------------------------------------------------- The only thing you need to change is n. Play around with it.

TY

There is a mistake at 38:53. It should be E[alpha] = alpha.

at 29:57 wouldn't the center of gravity of the pmf be (n+1)/2 not n/2 because the numerical values start at 0, not 1, so there are n+1 numerical values of uniform frequency. the middle would be (n+1)/2 no?

Why he kept calling excepted value mean? Mean is average,right?

I

Awesome series! However, starting at 31:00 he's using loose language and is not very precise with notation (or I am missing something). IMHO, it is not correct to write Y=g(X). Moreover, below that he is also writing g(x) (lower case "x"). I think the correct ought to be Y=g o X (where "o" is function composition). Only then does g(x) (lowercase "x") make sense. Otherwise if what he is writing "Y=g(X)" is the correct form, what is the domain of g?

if the gods give everyone a boost of 10cm... lolll we have a true champ here.

I thought "discrete" and "continuous" random variables were statements about their domains instead of codomains.

"god gives everybody a bonus of extra 10 cm"

20:36 Great

Greek professor!!

sir is explaning gud.. but slides are not visible properly.. can someone plz provide me it's soft copy?. it will be really helpful to me.. thank you.

43:44

which books do they follow??

177.5

Why you give as an examples about the hight , in min 40 , if we know that it is continuous variable and we talk about discrete variables ? Plz correct my ideas If I miss understand some thing

I believe in god now

NB

Can I become pretty good at probability after attending all the lectures of this course?

Se

Greeks. Russians. No difference. Just like with Nuclear Physics and Music Theory!

love this russian lecturer

so what ?

A "random variable is a function in programming".... mic drop!

2024 June ???

he is indeed very good but actually he is greek

I have seen a lot of videos on the internet mostly through RU-vid regarding random variables, binominal, polinominal and multinominal random distributions. I've seen several vids of this teacher and considering myself a guru on this particular subject, all I can say is that his approach, when it comes to teaching, is not the most adequate. To much theory and very few practical and clear examples. Being myself a college graduate and having the experience of knowing all lot of teachers, normally these kind of theorists are the ones who know less. They hide themselves in all this theory to try to disguise there lack of teaching capacity. Don't get me wrong, as theorists they're very good. Their lecture is very detailed and well studied. It's almost like you're hearing a recorder or a machine repeating the same thing over and over and over. Normally these kind of teachers won't even admit any sort of questions while the "recording" isn't finished playing (these are the worst kind). Even if you try to ask a question, anything out of the norm would cause discomfort, almost causing the "machine" to short-circuit. It's noticeable that he has a foreign accent. Maybe he doesn't feel very comfortable with his English and that's why he gives his lectures/classes the way he does. I don't know. I think I actually saw a video of this teacher in mutinominal distribution video where he actually says something like "the multinominal distribution is a NICE generalization of the binominal distribution"?!?!? "Nice generalization"?!?!?!? When your teaching or talking about maths there is no place for words like "NICE". When you're teaching mathematics try to be OBJECTIVE . No wonder I have seen several videos of this guy and almost not a single practical case with a final answer. There are a lot of good videos on the internet that actually give clear examples and with final results like this one ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-WWv0RUxDfbs.html

When talking about continuous random variables don't use the word random because there is nothing random about the distribution of the outcomes of these particular variables due to the nature of the samples being analyzed. Unfortunately this seems to be a misleading concept throughout almost all the videos on the internet regarding this subject. Even discrete random variables shouldn't initially be considered random due to the uncertainty of their distribution outcome. Uncertainty and randomness do not have the same meaning. An uncertain discrete variable could prove to be more likely random in it's outcome according to a more likely random type distribution. When analyzing continuous variable distributions you're are admitting the existence of a value range (similar to a discrete distribution), but contrary to a discrete distribution you are also admitting that, without any trials and without knowing the outcome, that you are going to have a certain type of distribution that follows a certain type of symmetrical curve. As the values tend to get further from the peak of the curve their frequency starts to decrease more or less proportionally. It's almost like a rule that you follow. Being that the uncertainty lies only in the variance of the distribution. Normally dealing with continuous distributions you are analyzing co-related values (correlation between a infinite value and it's frequency), who's relationship is given through a formula. The problem is that there are no predefined rules, formulas or predefined co-relations for uncertain outcomes. Notice that I used the word uncertain and not random.

his lectures are extremely trrible i don not u8nderstand how all comments agree that they are great it seems like they are watching kim during the lecture