 Hey guys, it's MJ the student-actree and we're going to be looking at the random variables of Tx and Kx And this forms part of chapter 7 survival models for subject CT4 So, yeah, let's jump straight into the material So we've got the random variables Tx and Kx. What exactly are they? Well, if we look at the definition of Tx, it is the complete future lifetime of a life-aged X And it is a continuous random variable This gives us a lot of information. So let's just unpack it. The first word we're seeing here is complete and This best way to think of this is We're gonna be looking later at Kx, which is the curtailed future lifetime or the incomplete future lifetime So what we mean by complete is we're counting everything So that's why this is a continuous random variable because if the future lifetime is 50 years six months three days and 22 seconds The Tx is going to capture all of that information is the complete future lifetime Also, let's look to the next word future. So this is something that We're gonna be making a prediction about we're gonna be forecasting And because it's into the future We don't know it for certain hence why it is a random variable So we don't know what the value is going to be and it can take on any number of different values and The value that we're looking at or why it's important. It is the lifetime of a life so Let's say you have a little puppy and you're like how much longer is my little puppy going to live? so if your puppy is one years old You want to know for how many more years? will you have your puppy for and Obviously the longer Or the greater the value the better it is because means the puppy survives longer So let's have a quick little Example regarding Tx and then I'm gonna come and talk about Kx So I've got a timeline and like I stress in all my videos time lines are very important They help you to visualize the problem so let's talk about I Actually did this I think when I was 22 so It was a little bit of an example of myself But yeah, let's pretend that this timeline is my age or my life span So at age 22 how much longer am I going to be living for and We don't know we don't know I could die tomorrow. So it could be the very next day I could you know, they could discover new technology and I live for another 200 years But let's say I unfortunately die at age 72 and a half It means that my complete future lifetime from the age of 22 would be 50 and a half And this is where it does get a little bit tricky is because we are looking at The difference between the start age and the end age. That is what the complete future lifetime is We're not worrying about the previous years that I survived We're only concerned with where I am now and where I'm going to be finishing And again, I want to just stress again that this is a random variable So this is going to have an expected value. This is going to have its own variance. We're not sure exactly where When I'm going to die Now let's look at something just a little bit simpler and that is KX and KX is the curtailed future lifetime of a life aged X and The big difference as is that this is a discrete random variable So whereas the continuous it could take any value Between you know natural numbers and whole numbers and stuff like that the discrete random variable can only take You know, it is a defined set. So What we're looking for here is we're just interested in the amount of years so in this case above We just that's the mathematical formula is we're going to be taking away the decimal point and we're just going to be looking at the whole completed years and Why this is why we'd want to look at the curtailed future lifetime It comes down to insurance and various annuity products and stuff like that in some instances With regards to pensions and retirements people get paid an amount at the beginning of every year so The reason why the insurance company would want you to be interested in KX and not TX is That they know that on my 72nd birthday. I'll get my lump sum I Don't get anything six months in I only get on my 73rd and my 74th And so they just want to know okay for this person. He's going to be living another 50 years They say retires at age 60 that means we're gonna have to give him 12 payments and that's where it comes in It's just a simplification It was done because back in the day they didn't have super advanced computers, but now with today's technology I mean you can see continuous random variables are getting more and more used over the curtailed discrete random variables but job that is The random variables of TX and KX key things to remember their random Variables, and you want to look at it's the time between the start point which is X and the end point Which is the random unknown point? And job These are going to be the other topics. We're going to be talking about so if you want to watch them as well subscribe to the channel or come back and check later and Yeah, hopefully they'll be uploaded by then. Thanks guys for watching and study hard. Cheers