1 00:00:00,120 --> 00:00:06,180 Now that we've talked about measures of central tendency like mean median and mode and measures of spread 2 00:00:06,180 --> 00:00:09,390 like range and IQ are interquartile range. 3 00:00:09,390 --> 00:00:16,290 We want to talk about this idea of transforming a random variable because if we have, for instance, 4 00:00:16,290 --> 00:00:22,830 a data set and we already know all of these measures about the data set, so let's just make up some 5 00:00:22,830 --> 00:00:23,370 values. 6 00:00:23,370 --> 00:00:29,550 Let's say that for this particular data set, we know that the mean is six, that the median is seven, 7 00:00:29,580 --> 00:00:35,100 the mode is three, and we'll just say the range in IQ R are ten and eight. 8 00:00:35,100 --> 00:00:38,280 And let's say we also just have some standard deviation. 9 00:00:38,280 --> 00:00:45,360 What we want to say now is that if we shift the data set by some constant value, we'll call it K, 10 00:00:45,690 --> 00:00:50,910 Then our shifted data set is only going to see the mean meeting and mode. 11 00:00:50,910 --> 00:00:56,820 B shifted the range and interquartile range and standard deviation for that matter will still stay the 12 00:00:56,820 --> 00:00:57,360 same. 13 00:00:57,480 --> 00:01:06,030 So if we shift the data set by some constant K, then the mean and let's say the shift is adding K to 14 00:01:06,030 --> 00:01:12,630 every value in the data set, then the mean of this new shifted data set will be six plus K. 15 00:01:12,660 --> 00:01:14,820 In other words, the mean will change. 16 00:01:14,820 --> 00:01:21,270 The median will also be seven plus K, the median will change and the mode will change. 17 00:01:21,270 --> 00:01:29,310 The mode will be three plus K, but the range and the interquartile range won't be affected by the shift 18 00:01:29,310 --> 00:01:31,440 of adding K to the data set. 19 00:01:31,440 --> 00:01:37,320 The range and interquartile range will still be ten and eight and the standard deviation is also not 20 00:01:37,320 --> 00:01:37,860 affected. 21 00:01:37,860 --> 00:01:42,840 So the standard deviation will still just be equivalent to the standard deviation from the original 22 00:01:42,840 --> 00:01:43,560 data set. 23 00:01:43,560 --> 00:01:50,190 And the reason for that, if we think about it, we could sketch a hypothetical probability distribution 24 00:01:50,190 --> 00:01:51,930 for the original data set. 25 00:01:51,930 --> 00:01:55,980 Let's say that the distribution for the original set looks something like this. 26 00:01:55,980 --> 00:02:02,130 So this is the probability distribution for this set of data, in which case we could maybe indicate 27 00:02:02,130 --> 00:02:05,670 here the mean and the standard deviation. 28 00:02:05,670 --> 00:02:09,419 So the mean in this case for this original set is six. 29 00:02:09,419 --> 00:02:15,870 And then we have our standard deviation here, shifting the data set by adding some value K to every 30 00:02:15,870 --> 00:02:22,470 point in the data set means that this whole distribution gets shifted to the right by K units. 31 00:02:22,470 --> 00:02:30,780 So we could visualize this shifted distribution here and we would say that the shift of K units is this 32 00:02:30,780 --> 00:02:31,530 distance right here. 33 00:02:31,530 --> 00:02:39,330 So this is a shift of K, and so we would have our new distribution and we would show there it's mean 34 00:02:39,330 --> 00:02:40,800 and standard deviation. 35 00:02:40,800 --> 00:02:46,380 In other words, let's say that some of the values in our original data set, we see that the mode is 36 00:02:46,380 --> 00:02:46,710 three. 37 00:02:46,710 --> 00:02:52,470 So we know that this original dataset has to include values of three, the median is seven. 38 00:02:52,470 --> 00:02:55,560 So the data set might include values of seven, it might not. 39 00:02:55,560 --> 00:03:01,890 But let's just say that some of the data points in this original data set are three, three, seven, 40 00:03:01,890 --> 00:03:05,640 ten, 12 and those are just a few examples. 41 00:03:05,640 --> 00:03:13,620 If we add some value of K to each data point in the data set, then three of course is going to become 42 00:03:14,040 --> 00:03:23,040 three plus K, So this will be three plus K, this is going to be seven plus K and we'll have ten plus 43 00:03:23,070 --> 00:03:30,900 K and here 12 plus K, And these then are the values in the new shifted dataset. 44 00:03:30,900 --> 00:03:37,920 So if we call this data set X and we call this data set Y, then these are the values in the dataset 45 00:03:37,920 --> 00:03:40,770 X and these are the values in the data set Y. 46 00:03:40,770 --> 00:03:46,560 If we knew that K was equal to, let's say, five, then we would know that these values were eight, 47 00:03:46,560 --> 00:03:49,020 eight, 12, 15 and 17. 48 00:03:49,020 --> 00:03:55,050 But just for some generic K, we can express the values in the shifted data set is three plus K, seven 49 00:03:55,050 --> 00:04:01,800 plus K, etc. And so practically that means that every value in our original data set, if we can think 50 00:04:01,800 --> 00:04:08,010 about maybe a data point here, maybe that's the value three in the original data set, maybe we have 51 00:04:08,010 --> 00:04:10,230 a data point here of seven. 52 00:04:10,230 --> 00:04:19,860 Then when we add K to that value, we're going to shift over to the corresponding value three plus K 53 00:04:20,160 --> 00:04:27,690 right here in the new data set, we're going to shift over a distance of K to this new value in the 54 00:04:27,690 --> 00:04:28,560 new data set. 55 00:04:28,560 --> 00:04:35,040 And so because of that shift, we can see how the mean is going to move from the mean in the original 56 00:04:35,040 --> 00:04:43,200 data set to that mean plus K to get to this mean in the second data set in the shifted data set, the 57 00:04:43,200 --> 00:04:48,840 median will also move because the whole data set moves over and the mode just represents a single point 58 00:04:48,840 --> 00:04:49,350 in the set. 59 00:04:49,350 --> 00:04:55,620 So if this was the point representing the mode right here, when we shift the data set over, of course 60 00:04:55,620 --> 00:04:57,390 that mode is going to shift as well. 61 00:04:57,390 --> 00:04:59,720 But you can imagine that the range and. 62 00:04:59,850 --> 00:05:04,950 The interquartile range won't change because the spread of the data doesn't move. 63 00:05:04,980 --> 00:05:11,730 The whole set just gets shifted over by the same amount, but the width of this probability distribution 64 00:05:11,730 --> 00:05:13,230 won't change at all. 65 00:05:13,230 --> 00:05:19,200 If the range here was ten and let's say the range of ten spans from the lowest value in the set, which 66 00:05:19,200 --> 00:05:22,230 maybe is two to the highest value, which may be is 12. 67 00:05:22,230 --> 00:05:23,820 And that's why we're getting a range of ten. 68 00:05:23,820 --> 00:05:27,060 Let's say that the minimum and maximum values in this set are two and 12. 69 00:05:27,060 --> 00:05:28,290 So we have a range of ten. 70 00:05:28,290 --> 00:05:36,510 If K is five, then two and 12 become seven and 17, then the range is still ten because the spread 71 00:05:36,510 --> 00:05:37,710 of the data is still the same. 72 00:05:37,710 --> 00:05:42,930 We just went from 2 to 12 over to 7 to 17 and the same with the IQ. 73 00:05:42,930 --> 00:05:43,080 R. 74 00:05:43,110 --> 00:05:45,000 The IQ r is also a measure of spread. 75 00:05:45,000 --> 00:05:50,760 It's not going to change because the width of the data distribution doesn't change when we shift the 76 00:05:50,760 --> 00:05:52,710 data set by some constant K. 77 00:05:52,740 --> 00:05:58,380 So that's the idea of shifting a data set, the idea of scaling a data set. 78 00:05:58,380 --> 00:06:03,750 So let's say we start with again this same data set with the same values. 79 00:06:03,750 --> 00:06:11,310 So the mean is six, median is seven, mode is three, the range is ten eight and some generic standard 80 00:06:11,310 --> 00:06:12,030 deviation. 81 00:06:12,180 --> 00:06:13,920 So we have our original data set. 82 00:06:13,950 --> 00:06:21,240 If we scale the data set, meaning instead of adding some positive constant K to create a shift, we're 83 00:06:21,240 --> 00:06:24,060 going to multiply by some constant k. 84 00:06:24,090 --> 00:06:30,870 For instance, if K is two, that means we're going to double every value in the original data set to 85 00:06:30,870 --> 00:06:32,370 get the scaled data set. 86 00:06:32,370 --> 00:06:39,930 If K is one half, that means we're going to divide in half or have every value in the original data 87 00:06:39,930 --> 00:06:40,320 set. 88 00:06:40,320 --> 00:06:48,270 So in that case then the scaled data set, if we're scaling by some value, K just like with the shift, 89 00:06:48,270 --> 00:06:51,240 the mean median and mode will all get scaled. 90 00:06:51,240 --> 00:06:57,720 So the mean now will be given by six K, the median will be given by seven K, the mode will be given 91 00:06:57,720 --> 00:06:58,860 by three K. 92 00:06:58,860 --> 00:07:06,030 But here's the difference now the range and the interquartile range are also affected by scaling, even 93 00:07:06,030 --> 00:07:07,680 though they aren't affected by shifting. 94 00:07:07,680 --> 00:07:14,400 So the range now will be ten K and the interquartile range will be eight K, If the original range in 95 00:07:14,400 --> 00:07:19,950 interquartile range were ten and eight, The standard deviation is also affected by the scaling. 96 00:07:19,950 --> 00:07:23,940 So the standard deviation will also need to be multiplied by K. 97 00:07:23,940 --> 00:07:26,550 And we can see why this scaling is true. 98 00:07:26,550 --> 00:07:31,530 Because if we took a super simple data set, let's say we started with a data set, there's just two 99 00:07:31,530 --> 00:07:34,200 data points which are five and six. 100 00:07:34,200 --> 00:07:40,560 So not the set described by these values here, but just a two data point data set, one values five, 101 00:07:40,560 --> 00:07:41,730 the others six. 102 00:07:41,730 --> 00:07:48,360 If we say that K is two and we scale up our data set by a factor of two, then our new data set is ten 103 00:07:48,360 --> 00:07:49,530 and 12. 104 00:07:49,620 --> 00:07:52,380 Well, the mean over here is five and one half. 105 00:07:52,380 --> 00:07:54,240 The mean over here is 11. 106 00:07:54,240 --> 00:07:59,430 So we can see that the mean also doubles in the same way that the actual data itself doubled. 107 00:07:59,430 --> 00:08:02,100 The median over here is five and a half. 108 00:08:02,100 --> 00:08:04,080 The median over here is 11. 109 00:08:04,080 --> 00:08:05,670 So that doubled as well. 110 00:08:05,730 --> 00:08:11,070 The mode there is no mode, but we can see that even if there was a mode. 111 00:08:11,070 --> 00:08:13,860 So let's say there was some other value here of five. 112 00:08:13,860 --> 00:08:15,180 Now the mode is five. 113 00:08:15,180 --> 00:08:21,210 The new data set would be ten, ten, 12, the mode would be ten in the scale data set compared to five 114 00:08:21,210 --> 00:08:24,000 in the original data set, it got doubled as well. 115 00:08:24,150 --> 00:08:26,610 But then here's where we see the range. 116 00:08:26,610 --> 00:08:32,760 So ignoring this again here, the range is one because six minus five is one. 117 00:08:32,760 --> 00:08:38,280 And when we double those values, the range doubles from 1 to 2 because 12 minus ten is two. 118 00:08:38,280 --> 00:08:39,590 So the range is affected. 119 00:08:39,630 --> 00:08:44,039 The interquartile range, we don't really have enough to calculate here, but we can see that if the 120 00:08:44,039 --> 00:08:48,510 range expands, then of course the interquartile range is also going to expand. 121 00:08:48,510 --> 00:08:50,850 And then our standard deviation, same thing. 122 00:08:50,850 --> 00:08:53,160 It's also a measure of spread. 123 00:08:53,190 --> 00:08:58,080 It's going to be affected by the scaling and we see that with the data distribution. 124 00:08:58,080 --> 00:09:03,120 So let's say that this is the distribution of our original data set. 125 00:09:03,300 --> 00:09:10,680 We could identify here the mean and then the standard deviation in this original set, scaling the data 126 00:09:10,680 --> 00:09:12,510 set by some factor. 127 00:09:12,510 --> 00:09:19,110 If K is greater than one, that means we're increasing all of the values in the set. 128 00:09:19,110 --> 00:09:24,270 And so what that's going to do is stretch out this distribution horizontally. 129 00:09:24,270 --> 00:09:29,220 It's almost as if we grab the right hand side of this distribution and we pull it to the right so that 130 00:09:29,220 --> 00:09:30,960 this whole thing stretches out. 131 00:09:30,990 --> 00:09:34,590 You can imagine as we stretch it out, it's going to get flatter. 132 00:09:34,590 --> 00:09:40,740 The vertical height of it will decrease because we have the same amount of area here that we're working 133 00:09:40,740 --> 00:09:43,020 with, the same amount of area under the curve. 134 00:09:43,290 --> 00:09:48,030 And if we stretch this out to the right, we're just distributing it over a larger interval, over a 135 00:09:48,030 --> 00:09:49,320 larger range. 136 00:09:49,320 --> 00:09:56,310 So that stretched out distribution might look something like this with its mean and standard deviation. 137 00:09:56,310 --> 00:09:59,550 And we can see here with this stretch, notice how this. 138 00:09:59,600 --> 00:10:04,200 Standard deviation itself actually gets stretched, this standard deviation. 139 00:10:04,220 --> 00:10:08,900 Here we have a shorter distance, whereas in the stretched data set, the standard deviation gets stretched 140 00:10:08,900 --> 00:10:09,330 out. 141 00:10:09,350 --> 00:10:16,460 We can see here too, if we think about this point here where the boundary of that single standard deviation 142 00:10:16,460 --> 00:10:23,790 intersects the curve and maybe the symmetric point right here, we could see that width. 143 00:10:23,810 --> 00:10:26,740 If we look at the same width along this curve. 144 00:10:26,750 --> 00:10:28,400 So maybe that's this point. 145 00:10:28,400 --> 00:10:36,380 And then roughly, let's say this point right here and we connect those, we can see how the width of 146 00:10:36,380 --> 00:10:42,050 just one standard deviation around the mean is larger than this width of one standard deviation around 147 00:10:42,050 --> 00:10:42,560 the mean. 148 00:10:42,560 --> 00:10:47,660 So everything about the spread, the range, the interquartile range and the standard deviation are 149 00:10:47,660 --> 00:10:48,620 going to increase. 150 00:10:48,620 --> 00:10:54,170 They're going to stretch out by whatever factor K is indicating if K is less than one. 151 00:10:54,200 --> 00:10:58,790 Of course, then this original distribution gets compressed even further, which means it's going to 152 00:10:58,790 --> 00:11:01,840 get taller instead of shorter. 153 00:11:01,850 --> 00:11:06,590 This way it's going to get taller and get compressed into a narrower range. 154 00:11:06,590 --> 00:11:12,260 But for values of K greater than one, we stretch out the distribution and all of those measures of 155 00:11:12,260 --> 00:11:14,410 spread get stretched out as well. 156 00:11:14,420 --> 00:11:21,860 So the takeaway here is just to realize that only mean median and mode are affected by a shift. 157 00:11:21,860 --> 00:11:28,280 But all of these measures mean median mode and range I.Q., R and standard deviation are all affected 158 00:11:28,280 --> 00:11:29,660 by scaling. 159 00:11:29,660 --> 00:11:35,540 So with that in mind, let's look at sort of a real world example so that we can understand how this 160 00:11:35,540 --> 00:11:37,190 affects data in real life. 161 00:11:37,490 --> 00:11:44,210 In this example, let's say that we're running a dropshipping business, so we're not the manufacturer 162 00:11:44,210 --> 00:11:46,250 who's actually creating the product. 163 00:11:46,250 --> 00:11:52,070 We have a separate manufacturer, they are producing the product, but then they send it to us and our 164 00:11:52,070 --> 00:11:54,740 responsibility is to just ship it out. 165 00:11:54,740 --> 00:11:59,930 So we are paying the manufacturer for the privilege of that shipping business. 166 00:11:59,930 --> 00:12:05,390 So we're getting paid by the manufacturer to ship out their product. 167 00:12:05,390 --> 00:12:11,090 But they're also asking us to pay a small upfront fee for the privilege of doing business with them 168 00:12:11,090 --> 00:12:14,600 and so that they're not sending us a bunch of product to ship. 169 00:12:14,600 --> 00:12:19,880 Then we fail to ship it and they get left with a bunch of us shipped product and lose a bunch of money. 170 00:12:19,880 --> 00:12:27,350 So the arrangement that we've agreed on here is that we will pay the manufacturer $2 per package or 171 00:12:27,350 --> 00:12:34,850 per item and they will pay us, or we will receive $10 for every pound of product that we ship out. 172 00:12:34,850 --> 00:12:40,370 So what we know about the product that we're getting from the manufacturer is that the items that they 173 00:12:40,370 --> 00:12:45,470 manufacture and the things that they want us to ship come in only three weights. 174 00:12:45,470 --> 00:12:50,000 There's a £1 product, a £2 product and a £5 product. 175 00:12:50,000 --> 00:12:56,690 So what we do is we create this discrete random variable X that models the weight of product. 176 00:12:56,690 --> 00:13:00,470 So X can only take on the values one, two or five. 177 00:13:00,470 --> 00:13:02,270 There are no in between values. 178 00:13:02,270 --> 00:13:05,990 There's no one and one half pound package, there's no £4 package. 179 00:13:05,990 --> 00:13:07,730 We only have the £1 package. 180 00:13:07,730 --> 00:13:11,330 And then maybe this is a two item bundle in a five item bundle. 181 00:13:11,330 --> 00:13:16,700 But the package weights that we get can only be a £1, £2 or £5 package. 182 00:13:16,700 --> 00:13:22,700 And these are the rates at which customers purchase these items or these bundles. 183 00:13:22,700 --> 00:13:32,660 So 60% of customers purchase the £1 item, £1 package, 30% of customers purchase the £2 bundle and 184 00:13:32,660 --> 00:13:39,170 10% of customers purchase the £5 bundle, which means that the manufacturer, when they are sending 185 00:13:39,170 --> 00:13:45,980 us product to ship out, 60% of what we're getting from them is these little £1 packages, 30% is the 186 00:13:45,980 --> 00:13:49,280 £2 packages and 10% is the £5 packages. 187 00:13:49,280 --> 00:13:55,010 So we can say the probability of us receiving from the manufacturer and shipping out to the customer. 188 00:13:55,010 --> 00:13:58,820 Any particular item here is given by these values. 189 00:13:58,820 --> 00:14:06,140 What we want to do is calculate a mean or an expected value and a standard deviation value for the amount 190 00:14:06,140 --> 00:14:10,340 of money we can expect to make based on this payment structure. 191 00:14:10,340 --> 00:14:16,940 So in order to do that, the first thing that we want to do is compute the expected value of the number 192 00:14:16,940 --> 00:14:22,910 of pounds were sending out to customers the weight of all the product that we're shipping to customers. 193 00:14:22,910 --> 00:14:29,990 So what we can say is that our expected value, the expected value of our variable x here, which models 194 00:14:29,990 --> 00:14:35,270 the weight of product, is going to be the mean of this variable here. 195 00:14:35,270 --> 00:14:40,490 And remember that when we have a discrete random variable like this, what we have to do is find the 196 00:14:40,490 --> 00:14:47,600 weighted mean, which means that we have to say for the £1 packages, there's a 60% chance that we're 197 00:14:47,600 --> 00:14:50,480 sending out one of those for the £2 packages. 198 00:14:50,480 --> 00:14:53,840 There's a 30% chance we're sending out one of those. 199 00:14:53,960 --> 00:14:59,450 And for the £5 packages, there's a 10% chance that we're shipping out. 200 00:14:59,630 --> 00:15:01,250 One of those packages. 201 00:15:01,250 --> 00:15:13,730 So when we multiply here, we get 0.6 plus 0.6 plus 0.5 or 1.7, which means that on average for every 202 00:15:13,730 --> 00:15:20,870 package we ship out, we're shipping out about £1.7 of product, even though the only values we're actually 203 00:15:20,870 --> 00:15:24,010 shipping out are £1, £2 and £5 packages. 204 00:15:24,020 --> 00:15:31,160 If we average out all the packages we're shipping, our mean weight is £1.7, then our variance. 205 00:15:32,010 --> 00:15:37,440 For this discrete random variable is going to be the difference between each of these weights one, 206 00:15:37,440 --> 00:15:39,270 two and five and the mean. 207 00:15:39,270 --> 00:15:41,760 So we'll say we'll take the first weight here. 208 00:15:41,760 --> 00:15:44,790 We'll say one minus the mean of 1.7. 209 00:15:44,790 --> 00:15:49,860 We square that value and we multiply it by the probability of that particular weight. 210 00:15:49,860 --> 00:15:57,330 So 0.6 and then we do that for all of the weights that we have, all of the values that X can take on. 211 00:15:57,330 --> 00:15:59,220 So we say here to. 212 00:15:59,890 --> 00:16:07,240 Minus the mean 1.7 quantity squared, and then multiply that by the probability 0.3. 213 00:16:07,390 --> 00:16:14,320 And then the last one is this last value x can take on five minus the mean of 1.7. 214 00:16:14,740 --> 00:16:19,870 We square that and multiply by the probability of a £5 package, 0.1. 215 00:16:20,020 --> 00:16:25,150 Now if we do that math, one -1.7 is -0.7. 216 00:16:25,150 --> 00:16:33,880 When we square that, we get positive 0.49 times 0.62 -1.7 is 0.3. 217 00:16:33,880 --> 00:16:36,730 When we square that, we get 0.09. 218 00:16:36,730 --> 00:16:44,560 So 0.09 times 0.3 and five -1.7 is 3.3. 219 00:16:44,560 --> 00:16:49,870 When we square that, we get 10.8, nine times 0.1. 220 00:16:49,870 --> 00:16:53,590 That's going to give us a value of 1.41. 221 00:16:53,590 --> 00:16:59,050 And when we take the square root of that to get standard deviation, we find a value of approximately 222 00:16:59,290 --> 00:17:00,550 1.19. 223 00:17:00,550 --> 00:17:07,720 So that's our standard deviation for this discrete random variable x modeling the weight of the amount 224 00:17:07,720 --> 00:17:09,099 of product we're shipping out. 225 00:17:09,099 --> 00:17:14,680 But remember, the whole idea here is that we want to figure out how much money we're making when we're 226 00:17:14,680 --> 00:17:21,130 getting paid per pound of product, but we're also sending the manufacturer $2 for every individual 227 00:17:21,130 --> 00:17:23,109 package that they give us to ship. 228 00:17:23,109 --> 00:17:31,990 So what we could do is we could set up a net gain equation, we'll call it here, capital N for net. 229 00:17:31,990 --> 00:17:33,790 We'll think about this as profit or loss. 230 00:17:33,790 --> 00:17:41,860 So net gain of our discrete random variable X and that's going to be equal to we'll write this as ten 231 00:17:42,100 --> 00:17:47,560 X because remember X models the number of pounds, it models the weight in pounds. 232 00:17:47,560 --> 00:17:55,060 So if X is pounds and we're getting paid $10 from the manufacturer to ship each pound, then the amount 233 00:17:55,060 --> 00:18:02,260 of money that we're going to make is given by ten X, except that we then have to subtract two from 234 00:18:02,260 --> 00:18:08,530 this net gain equation because we have to pay the manufacturer $2 for every product they give us. 235 00:18:08,530 --> 00:18:14,920 So if we then take this net gain equation and we evaluate it at all of the values that X can take on 236 00:18:14,920 --> 00:18:23,140 one, two and five we get when X is one, we get ten times one is ten, minus two is eight. 237 00:18:23,650 --> 00:18:32,140 When x is two for a £2 package we get ten times two is 20, minus two is 18. 238 00:18:32,530 --> 00:18:41,740 And for one of the £5 packages, when x is five, we get ten times five is 50, minus two is 48. 239 00:18:41,740 --> 00:18:51,610 So what this is saying then is that our net profit is $8, our net gain is $8 for every £1 package that 240 00:18:51,610 --> 00:18:58,420 we ship, which makes sense because it's a £1 package, the manufacturer is paying us $10 per pound 241 00:18:58,420 --> 00:19:02,350 that we ship, which means that we're going to get ten bucks from the manufacturer for shipping that 242 00:19:02,350 --> 00:19:03,340 £1 product. 243 00:19:03,340 --> 00:19:07,090 But we also had to pay them $2 for that individual product. 244 00:19:07,090 --> 00:19:13,840 So ten minus two, we get eight in contrast here, a £5 package because it's £5, they're paying us 245 00:19:13,840 --> 00:19:15,070 $10 per pound. 246 00:19:15,070 --> 00:19:21,820 We would make $50, except that for that one individual package, we paid them $2. 247 00:19:21,820 --> 00:19:25,270 So instead of making $50, we make $48. 248 00:19:25,270 --> 00:19:33,100 So now with these new values, we can set up a new probability distribution like we had over here instead 249 00:19:33,100 --> 00:19:36,070 of X and the probability of X. 250 00:19:36,100 --> 00:19:41,350 We'll look at this as N and the probability of N. 251 00:19:41,380 --> 00:19:48,430 In other words, we can think about N as a new discrete random variable that has been scaled and shifted. 252 00:19:48,430 --> 00:19:55,930 So the values that n can take on are 818 and 48 in the same way that X could take on the values of one, 253 00:19:55,930 --> 00:19:57,010 two and five. 254 00:19:57,220 --> 00:20:07,420 So we want to say here 818 and 48 and now we want to talk about the probability of each of these dollar 255 00:20:07,420 --> 00:20:08,080 amounts. 256 00:20:08,080 --> 00:20:11,200 And of course those probabilities are still going to be the same. 257 00:20:11,200 --> 00:20:13,630 The manufacturer is sending us product. 258 00:20:13,630 --> 00:20:17,350 60% of that product is still these £1 packages. 259 00:20:17,350 --> 00:20:21,550 So there's a 60% chance that we make $8. 260 00:20:21,550 --> 00:20:29,380 There's a 30% chance that we make $18 and a 10% chance that we make $48. 261 00:20:29,380 --> 00:20:35,770 So now if we want to find the mean and standard deviation of this new discrete random variable n we 262 00:20:35,770 --> 00:20:42,880 could of course, do the same thing we did before and use this distribution table to calculate the expected 263 00:20:42,880 --> 00:20:45,640 value and the standard deviation from scratch. 264 00:20:45,640 --> 00:20:52,990 Or we could use what we just learned about transforming random variables by shifting and scaling to 265 00:20:52,990 --> 00:20:58,630 just transform the mean and standard deviation we already calculated so. 266 00:20:58,690 --> 00:21:06,220 If you remember, we said that the mean was affected by shifting and it was also affected by scaling. 267 00:21:06,220 --> 00:21:10,270 What we can do then is say that our new mean. 268 00:21:10,270 --> 00:21:16,180 So instead of mu sub x, which represents the mean of the original discrete random variable, x will 269 00:21:16,180 --> 00:21:24,640 say that mu sub n the mean representing this new discrete random variable n has to be equal to remember 270 00:21:24,640 --> 00:21:32,740 we set up this equation for n in terms of x, so instead of ten times x minus two, we'll say ten times 271 00:21:32,740 --> 00:21:34,450 the original mean. 272 00:21:34,450 --> 00:21:37,330 So mu sub x and then minus two. 273 00:21:37,360 --> 00:21:43,720 And we include the ten and we include the minus two because the mean is affected by the scaling of ten 274 00:21:43,720 --> 00:21:46,180 and also by the shifting of two. 275 00:21:46,210 --> 00:21:49,810 So we can just plug 1.7. 276 00:21:51,110 --> 00:21:56,840 Into this equation and we get 17 minus two or 15. 277 00:21:57,170 --> 00:22:02,300 So the new mean, the mean here for this data distribution is 15. 278 00:22:02,300 --> 00:22:08,390 And of course we would find that same value if we used this same calculation we did before, where we 279 00:22:08,390 --> 00:22:20,690 take eight and multiply it by 0.6 plus 18 times 0.3 plus 48 times 0.1. 280 00:22:20,690 --> 00:22:22,640 So we could have calculated it that way. 281 00:22:22,640 --> 00:22:28,310 But instead of going through all that math, we can just scale and shift the mean we already found since 282 00:22:28,310 --> 00:22:33,380 n is just a shifted and scaled version of the original variable X. 283 00:22:33,380 --> 00:22:35,960 And then the same thing goes for the standard deviation. 284 00:22:35,960 --> 00:22:40,970 Except remember that standard deviation is only affected by scaling. 285 00:22:40,970 --> 00:22:43,250 It's not affected by shifting. 286 00:22:43,400 --> 00:22:50,480 So the standard deviation of this new discrete random variable n instead of plugging it in here to ten 287 00:22:50,480 --> 00:22:57,980 x minus two and replacing x with the standard deviation of x to say ten times the standard deviation 288 00:22:57,980 --> 00:23:04,700 of x minus two, we will ignore the minus two because the standard deviation isn't going to be affected 289 00:23:04,700 --> 00:23:06,770 by this shift of minus two. 290 00:23:06,770 --> 00:23:12,770 So the new standard deviation is simply ten times the old standard deviation. 291 00:23:12,770 --> 00:23:21,440 With respect to X, we know that that was 1.19, so this is approximately equal to ten times 1.19 or 292 00:23:21,440 --> 00:23:24,740 about 11.9. 293 00:23:24,740 --> 00:23:30,140 And of course, just like before, we could have found that the old fashioned way, we could have said 294 00:23:30,140 --> 00:23:39,140 here our data point eight minus the mean that we calculated 15 quantity squared times the probability 295 00:23:39,140 --> 00:23:43,880 of that particular value eight and then plus 18. 296 00:23:44,680 --> 00:23:49,570 Minus the mean of 15, square it, multiply it by the probability. 297 00:23:49,570 --> 00:23:51,700 And then the last one here, 48. 298 00:23:52,820 --> 00:23:54,710 Minus the mean of 15. 299 00:23:54,890 --> 00:23:57,440 Square it and multiply it by the probability. 300 00:23:57,440 --> 00:24:02,240 We could have done that whole calculation to get variance and then taking the square root of that variance 301 00:24:02,240 --> 00:24:03,800 to get standard deviation. 302 00:24:03,800 --> 00:24:07,340 And we would have found the same value here of approximately 11.9. 303 00:24:07,340 --> 00:24:14,600 But there's no need to go through all that work since n is just a data set under transformation by shifting 304 00:24:14,600 --> 00:24:17,660 and scaling from this original data set X. 305 00:24:17,660 --> 00:24:24,890 So whereas X modeled the weight that we were shipping out and models the amount of money that we make 306 00:24:24,890 --> 00:24:32,510 for doing that shipping and these values we calculate it for X tells us that on average the mean weight 307 00:24:32,510 --> 00:24:40,100 of each package that we ship is £1.7 with a standard deviation of £1.19. 308 00:24:40,100 --> 00:24:46,490 And then with that we can say that the mean amount of money we make for every package we ship out, 309 00:24:46,490 --> 00:24:50,390 we make on average a mean of $15. 310 00:24:50,390 --> 00:24:58,670 Even though for any given package, we will only ever make either $8 or $18 or $48 with no values in 311 00:24:58,670 --> 00:24:59,270 between. 312 00:24:59,270 --> 00:25:04,430 Our profit will only ever be one of these three values for an individual package. 313 00:25:04,430 --> 00:25:09,860 But when we take the mean there, when we look at the long term average, based on the fact that 60% 314 00:25:09,860 --> 00:25:16,340 of the packages we ship net us $8, that 30% of the packages we ship net US $18, and that 10% of the 315 00:25:16,340 --> 00:25:18,680 packages we ship net US $48. 316 00:25:18,680 --> 00:25:27,950 The mean amount of profit per package is $15 with a standard deviation of just under $12. 317 00:25:27,950 --> 00:25:34,310 And we were able to get all of this information here about the amount of money we make modeled by DN, 318 00:25:34,340 --> 00:25:40,730 Just by applying a scale and a shift to this X variable we already have. 319 00:25:40,730 --> 00:25:47,180 Instead of collecting all new data about the profit per packages and the probability of each of these 320 00:25:47,180 --> 00:25:53,960 amounts of profits and then calculating from scratch this mean here and this variance here, and then 321 00:25:53,960 --> 00:25:55,520 therefore standard deviation. 322 00:25:55,520 --> 00:26:02,120 So you can see how understanding this transformation of random variables by scaling and shifting can 323 00:26:02,120 --> 00:26:05,960 save us a lot of time answering real world questions.