What is a Random Variable?

Explains what a Random Variable is, using two examples, and also discusses the associated probability density function (pdf), and how it is modelled in practice.

Just one note to clarify the modelling Assumptions discussed in the video: Actually there are two parts to the assumptions being made. The first, is that there are three types of people who travel on trains: those who always use WiFi when travelling on a train, those who never use WiFi when travelling on a train, and those who use it on half of the trips they make (with a random selection as to which trips they use it on). Of course this is a simplification of reality, but that is what you need to do in order to build statistical models of reasonable/practical computational size. The second part of the assumption is that each train carriage will always contain two of the first type of people, one of the second type of people, and two of the third type of people. So, depending on what the two "third type people" have chosen to do, there will either be 2 people using WiFi (if both of the "third type" people have chosen not to use WiFi on that trip), 3 people using WiFi (if only one of the "third type" people has chosen to use WiFi on that trip), or 4 people using WiFi (if both of the "third type" people have chosen to use WiFi on that trip).

* And I should point out that I must have had a brief brain fade at the 4:53 mark in the video where I wrote X's instead of Y's. In other words, the probabilities should be written P(Y=0)=0, P(Y=1)=0, etc, instead of P(X=0)=0, P(X=1)=0, ...

[margaretblack]

I appreciate that you explicitly state that we know that P(heads) = P(tails) = 0.5 BECAUSE OF physics. I've never thought of that before, but it's really helpful to think of it in that way. There's always a reason behind even the most seemingly obvious things. Thank you!

[prasheenman]

for x=2 the first assumption that is 2/5 means 2 always uses so how you assumed it as 1/2

[hariharanramamurthy]

so sir are saying that giving numerical analogy to probability evnet is called random variable

[ZzSlumberzZ]

1:41 I'm probably nitpicking here, but I'd like to point out that the term probability *density* is reserved for continuous distributions. The analogous term used for discrete probability distributions is probability *mass* function.

This distinciton comes from the fact that conversion from discrete to continuous is not consistent mathematically (due to the extra dx term in continuous distributions.) For instance, we know that sum of the probabilities of the occurences of all possible values of random variables is 1. In the continuous case, the equivalent statement would be "area under a PDF curve is 1".
The formula for the total probability *density* function has the inner product of the form f_X(x)dx. Similarly for discrete random variable, it is just P_X(x). By comparing we can conclude, for mathematical consistency, that f_X(x) must have the units of "density" rather than "mass" (the analogy probably comes from linear density = mass / length). Thought this was interesting.

[najeebkhan]

Just one quick question, How can we associate probabilities with uniformly distributed random variables X= {0, 1}, in a string of bits. In other words, If I wanted to send a string of bits (that are generated uniformly) over some channel, then how can I get its respective probabilities that sums to one? A little help would be appreciated.

[arastooajorian]

I think pdf is a term for continuous random variables which shows likelihood for each point. So probability mass function is a more accurate term in discrete random variables

[vedantjoshi]

Hello sir, firstly thank you for such great videos.
I have a request though, can you make a video where you just put the formulas used to solve numericals so that students like me can relate the "explanation" to what is asked in exams!
Thanks!

[frederickrosas]

Sir, can you elaborate more why P(X=2)=1/4, P(X=3)=1/2, P(X=4)=1/4? I was lost on the use of assumptions. Thank you.

[abdullahikelly]

great explanation I've never thought of that before whether physics has something to do with heads and tails

[ayushkumarrai]

Sir, can you please make a video on Maximum Likelihood and Maximum A Posteriori decoding?

[edwinr]

Sir ...can you please make a video on DTFT, DFT and FFT

[GulzarAhmad-swkh]

shouldn't it be pmf instead of pdf?

[ziqijia]

Hello, sir, thank you for the explanation! But I highly suspect that you are an expert in chemistry according to your profile image. lol

[NK-juns]

When 2 people always use Wi-Fi, then p(x=2) should be 1 right? Why is it .25?