Random Variables and Probability Distributions

welcome back so today we're going to introduce a really really major Concept in probability the concept of a random variable so just like in uh algebra and

calculus you can have a variable X that takes on values two or three or Pi or whatever values a random variable in

probability and statistics is also a a variable that can take on uh a given set of values and then you can assign a

probability to each of those uh values of X okay so I'm going to give you some examples um in fact let's just start off with an example and then kind of Define

what we mean by X and give uh a more formal definition so um a good example is let's say that I am flipping My Fair

coin okay remember uh I have my My Fair coin here so I'm going to flip a coin four times flip a coin uh four

times and we've already talked about the sample space Omega of all of the things that could happen you could have heads heads heads heads or heads heads Heads

Tails Etc a random variable is going to be some uh real valued kind of um number that you associate to each of those

experiments in that sample space so let's say I you know flip a coin four times x my random variable could be the number of heads that occur this could be

the number of heads that occur so um we we know I'm actually just going to write this in Orange we know

our sample space um Omega for four um four coin flips is just I'm just going to list it out heads heads heads heads heads heads Heads Tails heads Heads

Tails heads uh dot dot dot dot all the way to taals taals Tails Tails okay um and I think there are uh 16 elements of

the sample space there's 16 different sequences of heads and tails I can flip um 2 to the^ 4 and this random variable

X is the number of heads that could uh occur in these coin flip sequences and

so X we would say we would say that X is uh takes values between 0 1 2 3 and four it could be an integer between 0 and

four I could get zero heads I could get four heads I could get one two three heads as well okay so X is a variable

and it has um you know this kind of um set of numbers that it could it could take so this is a discrete random variable my random variables could be

discret um they could be discreet like the number of heads in N coin

flips or they could be continuous they could be continuous random variables a good example of a continuous random variable would be the height of an average the height of an

American the height of Americans um height of Americans you could make it the height of Brazilians the height of uh French the

height of people worldwide it doesn't matter the height of Americans this could be in feet it could be in meters it could be in CET but this is a variable that could be

kind of any Continuum of values you know within some reasonable range discret random variables can only take on a discreet set of values okay now

importantly I can assign a probability to each of these values of X that's really really important so to each value of x I can assign a probability so the

probability of x uh equals Zer that I get no heads there's only one out of these 16 uh cases where I get zero heads so

this is a 1 and 16 probability um the probability that x equals 1 I'm just going to use shorthand um x equal 1 the probability um we're

going to use the binomial distribution we know that this satisfies or follows the binomial distribution remember Pascal's triangle uh and N choose R so

the probability that xal 1 is going to be uh I think 416 I think the probability that x equals

2 uh is 66 then x = 3 the probability again is 416 and the probability that x = 4

is 1116 so this just is a nice example uh of a binomial distribution and what we would say I'll get into this in a minute but

we would say x is distributed as a binomial random variable so it is a random variable that adheres to the binomial distribution of probabilities

um that we can calculate here and one of the cool things um that you can do then when you have this random variable and you can compute the probabilities of each of these states of the random

variable is this gives you a very nice concise way of plotting uh or summarizing those probabilities in a histogram in a plot so let's actually do

that so we're going to plot um the probability of X number of heads out of four coin flips so this is my variable

X um this is going to be 0 1 2 3 and four and I'm going to rough roughly try to do this it should go from 1 2 3 4

five six so the probability of getting zero uh heads the probability of X being

zero is 116th the probability of X being one of having one uh heads in this coin flip is

416 uh 1 two 3 4 16th each of these ticks is a 16th of the probability the probability of X equaling 2 is 616

that's this one here the probability of X equaling 3 is again 416 and the probability of X equaling 4 is 116th and

you should always label your axes this is 0 1 2 3 and four heads and this is

probability in 16 this is 1 16th okay good so this is a really nice concise way of summarizing this relativ

you know uh big set of things that could H happen all of these 16 possible cases of four coin flips can be summarized in this distribution this is called a

probability distribution over my random variable X now when I was learning Algebra I remember this idea of a variable being kind of a tough

abstraction I think this is something that kids um I had a hard time with this I remember my mom telling me she had a hard time with this um um you know my kids I remember there was a day or two

where they had a hard time with this idea of an abstraction of a variable so you're used to doing math with numbers 3 + 5 is 8 you know and then when you

start introducing x + 5 = 8 what is X that takes people some time to get used to now we're all super comfortable with that abstraction now but this might take

you a little while so it's okay if it takes you a minute for this abstraction to sync in and for you to have intuition we're going to do probably 10 lectures just on examples of random variables how

to compute functions of random variables what are some of the most common examples of Rand random variables and you're going to build this intuition just like you did in algebra and

calculus so it's going to be okay okay so um a random variable essentially is just like a normal variable except I can assign a probability to X being in each

of its possible states that that's really important um and maybe I'll write that down here so um we essentially have P of

x uh is a function you can plot is a function called a distribution called um it's called a pro a

distribution or a probability distribution um called a distribution and roughly speaking what

you need you need the values of X you need to Define kind of what is the the domain of this probability space and what is the range of this

probability distribution so the values that X can take and then you need to have um essentially the probability uh P of each of each value

okay probability p uh of each possible value of x so here we know that our our number of heads our random variable is the number of heads in four coin flips X

can take five different values and here are the probabilities of X taking each of those five values so we have defined a probability distribution over our random variable X and that's a really

really useful concept um this is the kind of thing that you're going to really be glad that you have access to this abstraction of probability um distributions for example let's say I

flip a coin uh a hundred times I don't want to list two to 100 possible coin flips it's it's uh it's like kind of incalculably large the number of

possible sequences of coin flips if I flip a coin a hundred times I don't want to ever write that down on a on a whiteboard um or try to enumerate that

but the number of heads that occur in a 100 coin flips is a binomial distribution it has a name and there is a function for the probability density

of each of you know for their being 20 heads or 21 heads or 22 heads and so to some extent you can write down the probability distribution sometimes we

call it a probability density function you can write down that probability distribution for much more complicated random variables for much more complicated processes where you could never enumerate all of the possibilities

so this becomes really useful um and we have this uh number of heads and N coin flips that is going to be uh we're going to say X

follows a binomial distribution we're going to Define what this is in one of the next lectures um it's related to those binomial coefficients um that we looked

at earlier if we have a continuous variable like the height of Americans This is actually going to uh in this case follow a normal distribution uh a

normal or a gausian distribution so you've seen this before it's kind of the bell curve a normal distribution here and you'll notice actually the binomial distribution if I have n gets very large

if I have 100 coin flips or a thousand coin flips this will start to approximate uh or converge to a normal distribution so really really kind of

useful stuff and you can do things like you can calculate what is the probability if this is my distribution of heights so this is x in let's say uh

feet and let's say this you know average height I don't know what the average height of Americans is let's say it's like five8 or something like that and maybe this is six feet and I don't know I'm just making up numbers right let's

say 5 six 7 4 3 two whatever you can compute the probability that someone is less than 6 feet tall you can compute

what is the probability that someone is less than six feet tall just by integrating all of these probabilities up to that point I can compute the probability that someone is between seven and 8T tall um you know by

Computing adding up all the probabilities between xal 7 and xals 8 so these random variables and these probability distributions on those random variables you can do all kinds of

things you can do calculus on those variables you can integrate probabilities and find what's the probability within some range that I'm within some range of values of X um I

can compute the expected value what do I expect if I just pick someone off the street what is the expected height of that person and how much spread

do I have in that expectation how would I be um surprised if someone was 2 feet shorter or taller than that expected value these distributions have all of

that information and you can Define functions on these random variables the probability is one function you can also Define uh the expected value of x or the

variance of X or the standard deviation of X and all kinds of other functions of this random variable okay good um so we're going to have a bunch more examples we're going to talk about

binomial normal Pon exponential gamma a lot of the most useful random variables that are useful for things like how long do you expect to wait at the DMV if

there's five people ahead of you and the average weight time is 2 minutes um how many emails do I expect to get in the next 10 minutes given a certain rate of

emails and would I be surprised if I got no emails in that time like these are the kinds of questions you can ask an answer now that we have this abstraction of random variables so maybe I'll just

write down some of the why so I've already said most of this but like why do we want random variables um first it's a

concise summary a concise expression of all of your probabilities a summary of all

probabilities good it's a concise of all of the probabilities um sometimes it's hard to count these probabilities remember if I have 100 coin flips I don't want to count all of the ways I

can get 50 heads my probability distribution over that random variable it's a function that I can write down so I don't have to count all of those

possibilities okay so sometimes we get a function for p ofx so we don't have to count so no

counting remember uh the older I get the harder it is for me to count I have to have everyone be quiet while I'm counting to 20 and so I don't like counting if I have a function for these probabilities I want to use that

function most of the time uh two the probability density function this probability distribution sometimes we're going to call it a PDF uh which is a

probability uh density function and usually I think of probability density functions for these continuous variables a PDF this PDF this

is just a Shand I'm going to use to define my probability density this PDF uh is a

model of a random process and this is a really really important idea here that the real world never is

exactly uh beri or binomial or normal or any of the distributions these are mathematical approximations just like if I throw ball you know and I write down f equals Ma I might neglect wind

resistance I might neglect rotational effects there are all these things these approximations I make to get my simple you know ballistic trajectory or my simple f equals ma descriptions or my

simple um you know Galileo's Tower of Pisa constant gravity like all of these things are are approximations that's also true in probability the PDF is just a model of a

random process um if I flip this coin really there's wind resistance probably it's not perfectly 50% probably the way I flip a coin might be like a tiny bit

bias there's all of these things that make it not perfectly uh random but this is a good model of that random process and an important point of that model is

that this allows you so we're talking about probability for the most part here but this dual notion of Statistics is let's say I collect data I should be able to test my hypothesis that my

system is by bomi distributed or normally distributed I should be able to test hypotheses this allows me to test

hypotheses uh with data so for example if I flip 10 coins in a row and I get heads all 10 times does that do I actually think that my

PDF is binomial um with you know equal probabilities that would be a hypothesis testing problem and I could get a probability of How likely that sequence

of 10 heads is given that my coin is binomial a fair binomial or brui coin and so you can do hypothesis testing you can also do parameter

estimation um parameter estimation uh again with data so this is a machine learning or a data statistics

problem if I know that my my uh distribution of Americans Heights is normal we know that the normal distribution has two parameters that completely define it the mean and the

standard deviation so I can take my data I can sample 100 people off the street and I can get a really good estimate of that mean and standard deviation and

then from that small sample I can say something important about the much larger uh distribution of you know

whatever 300 million people good okay um three you can visualize data in a much easier way you can visualize you can

visualize and compute so we can compute uh our probability density functions uh we can um we can plot maybe

I'll switch to Yellow we can uh compute things like the probability that X is less than 6 and greater than four let's

say you know what's the probability that a random person is greater than 4 feet and less than 6 feet this is something you can compute once you have this probability

distribution um I can compute how unlikely is event X how unlikely is it that I am greater than 7 feet tall all kinds of things like that uh and you can

do calculus um in this case this probability is going to be the integral from 4 to 6 of my probability density

function let's just call this P of X DX okay so you can do calculus on these continuous distributions if I had a disc

distribution I would be doing a sum you know from K = 4 to 6 of P of X you know equals k something like that okay you

can compute these things using sums or integrals on these distributions you can do computations with these um last thing and there's a lot more dot dot dot dot dot but the last thing I'm going to

mention is that you can build functions you can build uh functions uh on X you can compute the

distribution of x s that can be useful sometimes uh you can compute the distribution of X squ that's a function of X you can compute the expected value of x that's another function or the

variance of X that's a function um you can take an uncertainty a normal gausian distributed uncertainty and you can propagate it through some Engineering Process some manufacturing process or

some dynamical system I can take an uncertainty in my initial condition of a d of a differential equation and I can propagate that uncertainty and see how it spreads uh in chaotic systems it gets

spread around the chaotic attractor all kinds of interesting things you can build functions on X you can propagate X through dynamical systems uh you can propagate uh propagate

uncertainty um through Dynamics this is what gausian processes do um is they propagate uncertainty through Dynamics super super useful

ideas um okay that's probably all I want to show you right now um one thing I'll just very very briefly mention is that if we had this probability density

function that tells me the probability of X being at a certain value there is also this notion of a cumulative probability it's the probability of X

being less than a certain value um and that would just be the integral of this probability distribution up to that point x that's called the cumulative density function the CDF we'll talk about that later I just wanted to

mention that that's also a useful function of X is the cumulative distribution function okay um a lot more coming up soon I'm going to show you examples of binomial normal Pon

exponential gamma um you know a bunch more we're going to work out examples we're going to use these to compute really intuitive things that you're going to be able to use uh both in your

daily life and um you know to do better engineering okay thank you