Cumulative Distribution Function (1 of 3: Definition)

we've come far enough through continuous random variables that it is a good point to just start pause and look back a little bit and that will actually help us move forward with today's content so let's see how we can go in terms of all of the concepts and skills that we have covered so far in like the last couple of weeks think back to the very first lesson in this topic we had a look at we compared the statistics we did earlier in the year versus the statistics that we would spend this chunk of time on right what was the key difference between the old statistics we were looking at and the new ones a key very good so we called them discrete random variables right and that's where you've got you know things that you count like numbers of people or dollars things that are separated out versus discrete versus continuous random variables and this is what we have spent our timer but we needed to sort of point out that there are some real differences between them okay good so we looked at that and then we said with these continuous random variables that we would describe them in terms of the language we've had in like calculus land right we would use a particular kind of function to describe the chance of different outcomes and I continue to random arrow what kind of function was that Court has a probability density function well does so very good on these continuous random variables we said okay the name of the game is these particular functions here and we looked at their properties namely two of them what are the two properties that every probability density function has number one it all if you add up the area underneath it equals one and then there's the other one which we don't look at very much but yeah the probabilities have to be positive right or zeros fine but you're kind of negative probabilities okay we'll come back to this guy in a second we then have a look at the simplest kind of probability density function the very simple kind started with a you we call them uniform probability distributions and then of course all the rest of them that are not uniform so we looked briefly at that and then we've spent most of our time I'm working with all these probability density functions and using them to find different things so for example we might use the probability density function as the name suggest to find out a probability right you'd go to the function you'd integrate from whatever lower boundary to whatever upper boundary and that would give you a probability what other stuff can we find with the probability density function I'll give you a clue they all start with M ok very good so so we call these measures of central tendency very good and they all start with M so we had our first one the easiest one was the mode that's just the highest point on your probe a density function yeah and then we said median what's the mean how do we find the median on a probability density function it's the middle right so half of your scores or whatever are in one side of the median are from the other and then the the most recent we did was main what we also call me average in this context start with a we call it expected value right so it gets this different name in a probability density a probability distribution I should say on one more thing just because it's on the board what what are our abbreviations for median and mean what are the okay so we do use x-bar sometimes but more frequently we've been using another Greek letter so it's like a you with an extra bit on it then think it's it's mu right its mu mu and then what that median median okay so we're coming to I'll come to that in a second we generally use q2 which is a bit weird right you're like meaning why is it q2 it's actually there's a good reason for what's the Q stand for it's the quartile and it's the second one right okay um I left off one set of brackets here right from from these guys sorry well you guys have done a lot of work right we would then say okay you know where the mean is you know where the mean is now I want to know what's like how much do you spread out from the mean remember this okay so we said from these there's this big idea of variance from which you can also get you all seen this to me before from which you can also get standard deviation and that standard deviation is why we use Sigma s fur standard okay so this is where we have been so far just what I want you to notice is as you look at all of these different concepts I want you to notice how frequently to work out any of these things how frequently we need to take this probability density function and then basically we integrate right like do you notice like every second question or or more frequently actually you're like Oh take it and then integrate take it and integrate there's probably one notable exception which is this guy the mode I'm you know how we're looking for the highest point on the probability density function you're looking for the maximum of this so instead integrating you might differentiate or you might just look at the end points okay but for everything else pretty much we are integrating okay which is why the process of integrating this guy the probability density function when you integrate it because this hop camp camp happens comes up and happens so frequently it gets its own name and that's the first concept we're going to have a look at today so the heading you can make is cumulative distribution [Applause] close function if you integrate a function you're going to get another function right so this is called the d/f PDF integrates up into the CDA so this is our this is our heading for today okay now let's uh before we start looking at like the notation for it I want us to understand what on earth it means so firstly I've told you it's basically what happens when you take your density function and then you integrate but I want to understand the name like what's what's this about okay cumulative so let's think about frequency which is a an idea we got from our discrete random variables versus cumulative frequency because this is where we usually use the word cumulative right okay so I'm sorry I keep doing this to you but it's just an easy example that I think you can all attach to let's think one more time about Heights in this class okay so I might say what is the frequency I know sorry Zacky I'll pick a different height this time um what's the frequency of max how tall are you can you tell us your actual height you know 190 far okay so 590 right so if I asked stay with me atop what is the frequency frequency of heights of 190 centimeters or kind of like a hundred and ninety right in this class and the answer would be one right that would be the frequency but if I asked what's the cumulative frequency of that same height what that would mean is I include that height but then I also include everyone before that right does that make sense so we we accumulate so in this class that would be oh is that twenty this no nineteen twenty today what is it today today six okay so you can see what that does is it includes Max's height and everyone beneath Max's high right well before okay so you accumulate everything so now I want you to think about what that might mean here okay so what is the probability density function give you it gives you a probability right of a certain event taking on a certain value by the way um we've talked we've used this notation before but I remember I think it's germane actually asked me this question she's like what's with the big X and the little X so if this is something that's just gone over your head this is as good a point as any to highlight what it means that big X is what thing are you interested in measuring right now what's what's the variable okay so in the example that I just gave you its height but it might be temperature or it might be population size or something like that okay so this is what's the actual variable being measured and then this guy over here the little X is what value are you interested in that variable taking for example 100 M 90 centimeters that's a value the variable is like the height okay so this gives us the probability of a certain thing but if I must say okay in a cumulative way I'm now asking not what's the probability of a certain one thing happening but what is the probability of that thing happening or any height less than that remember that so how would I know take this well it's it's anything less than or equal to that particular number that you're interested in that particular value so make sense and that'll be the little X okay so there's the value there's the variable okay so what we do is we say this is our definition okay our cumulative distribution function is or equals rather the probability of being less than some certain thing okay now you might remember when we were first introduced to probability density functions we don't like having these one-sided inequalities because when we're integrating because that's that's what we're going to do eventually you've got to start somewhere you got to end somewhere does that make sense right so instead of saying it this way what we'll do is we'll say you start from wherever wherever your values start your lowest height your lowest temperature we generally call that what the lowest thing is it zero I mean sometimes it's zero but we generally use a 1/3 like beginning right do I go from A to B A to B so I'll start at a and then I will go up to and include whatever value you're interested in right now does that make sense okay yeah yes that's exactly what it is but but in a specific like for acidity value yeah exactly okay now once you get to here I just put your pens now for a second because I'm going to show you something to not write which is that weird I do so pens now once you've gotten to this point okay now we normally are pretty okay with writing the integral that comes from here we would normally say you integrate from start to end like so and then we would normally say oh well you're integrating the probability density function which we usually call what like what's the typical name of it it's just f of X right so don't write this place you would normally say putting in your probability density function and then integrate okay and the reason I'm telling you not to write this is I wonder if you notice something kind of peculiar about what I've just written you integrate this thing with X and then once you've got a primitive you then put your lower and upper boundaries in right yes I know I know but you're going to acute it from the start to whatever value you're interested in okay but but this is the thing which is weird right you're like wait a second you're integrating with respect to X but then the thing you put in is X that's a little bit weird right so now this is what I would like you to write instead of writing f of X DX we write f of T DT now this is a very minor difference in the end we basically do the same thing and I'm gonna give you an example in a second right whatever your probability density function is you just pop it in integrate it and then substitute in your a and your X but just as a point of formal notation we don't like putting X into X so we just give this a different name okay

Channel: Eddie Woo

Views: 22,117

Keywords: math, maths, mathematics

Id: ZJsOOCghQJ0

Length: 12min 34sec (754 seconds)

Published: Thu Oct 01 2020