1 00:00:03,380 --> 00:00:08,810 When we are investigating probability density functions, it is very useful to be able to describe them 2 00:00:08,810 --> 00:00:14,390 using a few key properties and these properties can be the main, the variance and the skew. 3 00:00:15,320 --> 00:00:20,750 Now, we've already looked at using the expectation of radar to calculate the main of the distribution, 4 00:00:21,020 --> 00:00:25,030 but it can also be used to calculate these other useful properties as well. 5 00:00:26,340 --> 00:00:32,310 He we have a probability density function, and you can see from the distribution that most of the likelihood 6 00:00:32,310 --> 00:00:34,690 area is in this area here. 7 00:00:35,130 --> 00:00:40,440 So if we were to calculate the mean of the distribution will end up with a man looking like this. 8 00:00:41,100 --> 00:00:47,040 And in the past, we've shown that we can use the expectation of radar here to calculate the mean of 9 00:00:47,040 --> 00:00:47,940 the distribution. 10 00:00:49,430 --> 00:00:55,190 Another useful way to describe a distribution is by its variance, so this distribution here can be 11 00:00:55,190 --> 00:00:59,120 said to have quite a high variance compared to the second distribution here. 12 00:00:59,450 --> 00:01:03,230 So this distribution here is a low variance because it's more squashed in. 13 00:01:03,740 --> 00:01:09,100 The variance is a measure of how far away the distribution spread away from the main. 14 00:01:10,340 --> 00:01:14,540 The variance of the distribution can be calculated using this equation up here. 15 00:01:14,960 --> 00:01:20,360 So this equation here basically takes the difference between the random variable and its main squares 16 00:01:20,360 --> 00:01:23,820 it and works out the main value of that whole term. 17 00:01:24,440 --> 00:01:26,750 So it's a measure of the spread of the distribution. 18 00:01:27,440 --> 00:01:34,470 Now, this equation can also be rewritten b b range by expanding it out and written in this form here. 19 00:01:34,480 --> 00:01:40,700 So you can basically say we're working out the expected value of the random variable squared minus the 20 00:01:40,700 --> 00:01:41,400 main square. 21 00:01:41,420 --> 00:01:43,760 It's going to give you the variance. 22 00:01:44,540 --> 00:01:48,810 So the variance is equal to the standard deviation squared. 23 00:01:48,830 --> 00:01:55,290 So when we have a square here, we can we can say it's a variance if it's just the single sigma here. 24 00:01:55,580 --> 00:01:57,420 This is just a standard deviation. 25 00:01:58,910 --> 00:02:04,670 So using the mean and the variance, we can describe a distribution and it's fairly common to write 26 00:02:04,670 --> 00:02:06,990 a distribution such as like this. 27 00:02:07,310 --> 00:02:10,640 So here this just means that we have X, which is a random variable. 28 00:02:11,300 --> 00:02:18,050 The distribution has a mean of exposure and a variance of this sigma squared here. 29 00:02:20,270 --> 00:02:23,880 The last property that we're going to look at is the skewness of the distribution. 30 00:02:23,900 --> 00:02:28,670 So this is also another useful way of describing the shape of the distribution. 31 00:02:29,390 --> 00:02:34,640 So, again, looking at this distribution here, this is a fairly symmetric distribution. 32 00:02:34,650 --> 00:02:36,040 So we have the main value here. 33 00:02:36,050 --> 00:02:37,910 And you can see that on both sides of it. 34 00:02:38,210 --> 00:02:40,210 It's pretty much the same shape, just married. 35 00:02:40,220 --> 00:02:42,560 So it is a very symmetrical distribution. 36 00:02:43,340 --> 00:02:48,980 If we can compare this distribution to this distribution chain here, you can see that it's no longer 37 00:02:48,980 --> 00:02:49,670 symmetrical. 38 00:02:49,940 --> 00:02:53,030 So now it's a lot more asymmetrical on one side than the other. 39 00:02:53,690 --> 00:02:58,460 So the Skewness is a measure of how asymmetrical the distribution is. 40 00:02:59,300 --> 00:03:04,160 More specifically, the skewness of a distribution is a measure of the asymmetry of the distribution 41 00:03:04,160 --> 00:03:04,860 from the main. 42 00:03:05,330 --> 00:03:08,220 And again, it can be calculated using this equation here. 43 00:03:08,810 --> 00:03:16,130 So instead of the variance, we taking the difference from the main and we're doing it to the cube instead 44 00:03:16,130 --> 00:03:16,670 of the square. 45 00:03:17,330 --> 00:03:19,970 So this is going to give us the skewness of the distribution. 46 00:03:20,780 --> 00:03:26,060 Now, this year, the skew is also very difficult to compare between distributions. 47 00:03:26,090 --> 00:03:32,570 So what we normally do is that we normally calculate the coefficient of Skewness, which is just equal 48 00:03:32,570 --> 00:03:38,750 to the ski value divided by the standard deviation of the distribution to the power of three. 49 00:03:41,590 --> 00:03:47,080 So to recap, the three different properties that are usually used to describe the distribution are 50 00:03:47,080 --> 00:03:50,440 the main, which can be be considered as the first moment. 51 00:03:50,770 --> 00:03:55,400 The variance, which can be considered as the second moment, as you can see here, is to the power 52 00:03:55,400 --> 00:04:01,030 of two and the SKU, which is usually considered the third moment because we have the DePaolo three. 53 00:04:01,870 --> 00:04:06,820 Now, there's also higher moments, but they become a bit more abstract and not as useful at describing 54 00:04:06,820 --> 00:04:08,470 a probability distribution.