 Hi everyone, it's MJ and in this video we're going to be looking at a sample question from Exam P and it says the amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of D, the insurance company reduces the expected claim payment by 10%. Calculate the percentage reduction on the variance of the claim payment. Before we start, let's just say the following. Let X be before the deductible and let Y be after the deductible. Because at first this seems like quite a difficult question but if we go through the steps quite slowly, we're going to see that it's actually not too bad. Also let's say let Y, sorry, let Lambda be the mean of X. So we just want to use this parameter to be the mean of X instead of how we sometimes use it as 1 divided by Lambda. We're not using that, we're just going to say Lambda is the mean. Which means the expected value of X is equal to Lambda and because we're dealing with the exponential distribution it means the variance of X is going to be equal to Lambda squared which means the expected value of X squared is going to be equal to the variance of X plus E of X squared over there which is equal to 2 Lambda squared. We want this in order to work back to variance when we start using the reduction because we can't really just do the reduction straight on the variance because things get a little bit messy if we approach it from that mathematical way. Because what we see is the expected value of Y, which is the claims after the deductible is being reduced by 10% so it's 0.9 Lambda. Now we can't just say variance of Y and jump straight into it. What we have to do is we have to say what is the expected value of Y squared and in this situation it is going to be equal to 0.9 2 Lambda squared over there which we're getting from over there which we worked out a little bit earlier and that gives us 1.8 Lambda squared. Which means now we can do the variance of Y because it's going to be the expected value of Y squared minus the expected value of Y squared like that. Which means we have 1.8 Lambda squared minus 0.9 Lambda squared because we're getting it from over there and this is going to be equal to 0.99 Lambda squared and we're going to see that this compares to this and the difference between the variance of X and the variance of Y is 1% and let's see is that a potential answer and it is. So there we go. So for this question we did have to use something known as the memoryless property but remember the big trick here is use your expected values and if you want to calculate your variance. But if you've got any questions let me know in the comments section and I'll do my best to answer any queries that you have. Thanks guys for watching. Cheers.