What is a Probability Density Function (pdf)?

okay so for probability density functions I find it easiest to start by thinking about an example so let's think about the time it takes someone to drive from Los Angeles to San Francisco for example so this if we plot this we know that and this is the time on this axis here and let's say we just say the number of people who drive on a particular day well we know that nobody's going to take negative time and probably if someone is breaking the speed limit they might take five hours so there'll be some people who take five hours many more people will take six hours and probably some people will take a break and take seven hours more of them I expect also some people will take a couple of hours break and take eight hours less people will be taking nine and probably not many you take ten so this is a if we map it in terms of whole hours then we'd have a histogram like this where these are the number of people on that given day well what if we so this is the number what if we take the number and divide by the total then all it does is to scale this axis and it gives us a percentage of the total and now what if we were to say that we were going to not just do it on a particular day but do it over every day of the year and let's say we're not just going to take single integer hours but let's say we're going to measure the time with seconds precision or maybe even more accurate than seven seconds precision may be microseconds then we would have a curve that would be filled in if it wasn't done on just whole hours but just done with in placee infinite precision in time we would have a curve that's filled in and this is a function and so this is where we have this probability density function idea first coming here now in some cases if the system is a go deck which we might go into just here but in general many systems are then this is the same as what you've got for a probability density function so let's say you have a random variable X then you're going to have a probability density function now some people use little F for the probability density function others use little P I've prefer to use little F now let's say your random variable is capital X in this case X was the note that the time it takes to travel between Los Angeles and San Francisco we had that as T but in general that's used the x for the random variable and we denote that with a capital X as a subscript on our function this is a function of something it's a function of the values that X can take so in this case you often use little X in brackets and this can be confusing notation but let me just explain it again we use little F to indicate that it's a probability density function so that's little F to remind you it's a function the subscript tells you it's the probability density function for this random variable X in this case that random variable is the time to drive between San Francisco and Los Angeles for example but there's all sorts of other values all sorts of other random variables X could be the height of people a height of a person sitting on a train it could be the length of the person's feet as they come in you know whoever comes into a issue shop what's the length of their feet it could be the maximum temperature on the 1st of March anything that's random is how then and has a numerical value that is a random variable and so we're just going to use X to indicate whichever random variable is we're looking at you might have why there's a different ones there just a different one however you label them and the capital is the subscript to tell you that's the probability density function for that random variable and the thing in the brackets are simply what you plot against so we could equally change the little to alpha here and call this plot with respect to alpha and change that to alpha it would still be the probability density function for the random variable X is just you're using a different variable to plot it and I think it's important to make that point to hopefully explain what this is the capital X tells you it's that the PDF probability density function for the random variable X the thing in the brackets is just what you're plotting it against and so in general this can have a shape any sort of shape in this case the one I've drawn here it has a possibility of having negative values over here the time taken to drive between two cities obviously never negative in this case other random variables could have negative values and it could have a shape like this okay now what exactly is this for one thing we know about this is that the total area I mean what exactly is the PDF what exactly is this curve telling us well one thing we know is that the area under here equals one it's a bit like this percentage here all of them have to add up to one so this area equals one and in mathematics that's an integral from negative infinity to infinity of F X X or alpha whichever one we like to put D alpha or it could be F X there people little X in here DX okay I'm just putting the Alpha to show you that it's just the thing that we are plotting it against so it's just the thing that we're integrating against okay the capital X is the thing that tells you it's the PDF for that random variable and so this total area equals one this is something that we know and I think this helps me to understand what this actual height of this curve actually is like what does that number there actually mean what is that actually telling me well because it's in once you go to infinite precision the values of alpha here could be anything at all it is any value with infinite precision so actually the exact probability of getting any of these values of alpha exactly is zero it's kind of interesting concept when you think about it if got infinite number of possible values that your random variable could take then there's with infinite values that could possibly take then the probability that getting any one of those values has got to be very very small it's actually going to be zero because if there's infinite number of those values so therefore we actually it's better to think of this as a as a an area here a very very thin very very small area and hopefully now you'll see why I labeled this alpha because I'm going to call this I'm going to pick this value X so if I pick that value X and a little tiny little place next to it a little distance along X plus DX so where DX is very very small then the distance along there is DX it's a very very small distance and the height of this is going to be FX of X that actual height for that value of x the height is FX of X and we know and by the definition this is I think the way to the best way to understand the probability density function if we wanted to know the probability that our random variable is bigger than little X it's bigger than that value it's in is that in this range smaller than X plus DX so what's the probability if we did a random sample of the cars we pick a car at random and we say how long did it take that particular car to drive between San Francisco and Los Angeles so if we randomly pick a car and we ask ourselves what's the probability that the time that that car took was bigger than middle X and smaller than little X plus DX I was in this range then that equals the area of this this shaded area that I've drawn so it equals the distance along the bottom the distance along the bottom is DX times the height and the height was F X of X ok so this is actually the definition of the probability density function okay so it's a it's a function it has the property that for each little tiny segment the if you take the the the base times the height of that tiny rectangle and infinitely small it's a rectangle because these two values become together when this is infinitely small so the height times the base of that rectangle tells you the probability that your random variable is in that range that's the formal definition of their probability density function of course once you see it like this you can see that the integral is adding up all of those probabilities because of that is this function here that I've just said and if you add them all up you equal one so don't forget to like and share this video and subscribe to the channel it helps with sharing the the descriptions for other other students other people are interested

Channel: Iain Explains Signals, Systems, and Digital Comms

Views: 29,553

Keywords: pdf, p.d.f., probability, density function

Id: jUFbY5u-DMs

Length: 9min 45sec (585 seconds)

Published: Wed Feb 12 2020