 Hi everyone, it's MJ the Fellow Actuary and in this video we're going to be looking at the mean of compound distributions. So this is a famous exam question. It says let N be the number of claims on a risk in one year. Suppose claims as X1, X2 going forward are independent identically distributed random variables, independent of N. Let S be the total amount claimed in one year. We'll have the expected value of S and the variance of S in terms of the mean and variance of N and X. So what we're going to be doing in this video is just looking at the expected value of S. In following videos we will then look at the variance of S as well as the moment generating functions and how they apply to the Poisson distribution binomial. So yeah, like I said in the last video, compound distributions, they're part of Ruin Theory. This is where they're fitting in the actual world. However, you don't have to worry about that now because we are going to go into some of the abstract maths and we're going to be looking at deriving some of their moments. So our exam question, we want to take the expected value of S and we want to express it in terms of the X's and in terms of the N. So let's go through it. So first off we have S is equal to X1 plus X2 plus dot dot dot dot plus XN. Now remember N is a random variable. It could be 3, it could be 10, it could be 100, we don't know. But what we do know is that the expected value of S is equal to the expected value of the S given what N is. So we kind of get this next expression over here. Now for those of you who have the actuarial orange book, they will give you this formula. This is a formula that you can take into the exam. However, it's quite intuitive. I mean, what is it? My total losses? My total losses depends on how many losses occurred. So that is the logic. It's quite nice to follow when we come to the variance of total loss. You'll see that it's quite difficult to see that. But make sure that you're comfortable with that. The expected value of my total losses is equal to my expected value of my total losses when I know what the number of losses is. So like I say, that is thinking this expression through. However, it is provided for you in the orange book. Okay, so now that we have this expression, what we can do is we can expand the S into our expression of our X's. So that's simply all that's all we're doing in this next step here. Replacing S with its expression given by the X's. Now what we do is we can break this up into each of these things having their own expected value. And what's nice here is that the expected value of each of the total of each of the losses does depend on N on the number of losses. So the first loss does depend on the number of losses because think about it. If our total number of losses is 100, then the 101st loss will be 0 because it didn't occur. The 103rd one will be 0. So this is the lovely thing about why we can say given N, we don't have to say plus expected value of XN plus 1 given N because all of these terms going here will simply be 0. So what's nice about this expression here is that we're limiting it up just towards XN. And because if we've given what N is, we can kind of contain what this randomness is going to be. Then what we're doing is because like the question said, the X's are independently and identically distributed, then what we can do is we can add all of these together. And we know that there's going to be an N amount of them. We know it's N because like I said, N plus 1 is going to just fall away. It's going to be 0. So we can contain it to N. Also, we can add these things. We can break it out like that due to the expectation function, which like I say, you would have covered in the earlier actuarial statistics subject. So if you're getting confused here or actually confused anywhere here, like I say, go back to that actuarial statistics course. There's 12 hours that goes through all of this information. Big thing to understand here though is our expected value of the X's is now a constant. That is now a constant and the N is the random variable. So because of that, we can now take the expected value of X out. This is also again how expectations work. Remember, if you multiply it by a constant, you can simply take the constant out. It's not as straightforward when it comes to variance, but with expected values, you can simply take it out. And that's what we're doing here. So if you get confused coming from this step to this step, it's important to realize that the expected value of X is a constant. And our N is the random variable over there. Of course, expected value of N is now no longer a random variable. It is now a constant as well. So what we have is that what we're basically saying is that the total losses is equal, the expected total losses is equal to the expected number of losses times the expected loss of each one. And this is again quite nice and intuitive is because you can think of it as the total claims is equal to the expected frequency times the expected severity. And one way we measure risk is by saying it's frequency times severity. So that's very, very nice how it all kind of comes together. It is very intuitive. In the next video, we are going to be looking at the variance of S. And we're going to see that, okay, there it does get a little bit more tricky. It's not as intuitive, but we're going to do that in the next video. And I'm going to see you guys there. Thanks so much for watching. Cheers.