1 00:00:00,090 --> 00:00:07,170 We recently finished talking about how to transform a random variable by shifting it or scaling it by 2 00:00:07,170 --> 00:00:08,430 some constant. 3 00:00:08,430 --> 00:00:13,970 But now we want to talk about how to combine multiple random variables together. 4 00:00:13,980 --> 00:00:20,670 It's a similar concept in the sense that we are applying a kind of transformation to the random variables 5 00:00:20,670 --> 00:00:29,310 by combining them, but instead of applying a simple shift or scale to one random variable, here we 6 00:00:29,310 --> 00:00:36,060 have two different random variables, and we're combining them together as either a sum or a difference. 7 00:00:36,060 --> 00:00:39,150 And this has all kinds of real world applications. 8 00:00:39,150 --> 00:00:44,190 If we know how to do this, we can save ourselves some time, especially if we already have particular 9 00:00:44,190 --> 00:00:47,580 statistics about the individual random variables. 10 00:00:47,790 --> 00:00:52,860 So what we're saying here, let's go through an example as we do this so we can see what's actually 11 00:00:52,860 --> 00:00:53,580 going on. 12 00:00:53,820 --> 00:01:00,270 Let's say that we work for a company and we're trying to get an understanding of the total time that 13 00:01:00,270 --> 00:01:04,349 it takes for our company to get our employees paid. 14 00:01:04,349 --> 00:01:09,150 And we realize that what we really have here is two random variables. 15 00:01:09,150 --> 00:01:17,620 We have a set of data about how long it takes the manager of each department to approve the time cards 16 00:01:17,620 --> 00:01:19,440 of the timesheets of our employees. 17 00:01:19,620 --> 00:01:25,890 And then we have a separate set of data about how long it takes our payroll department to process those 18 00:01:25,890 --> 00:01:28,020 approved time cards or timesheets. 19 00:01:28,020 --> 00:01:33,690 But let's say that these two activities in total constitute all of the time that the company spends 20 00:01:33,690 --> 00:01:38,010 to actually get over the finish line of issuing paychecks to our employees. 21 00:01:38,010 --> 00:01:45,330 What we want to do is think about this set of data as the random variable x and this set of data as 22 00:01:45,330 --> 00:01:47,100 the random variable Y. 23 00:01:47,130 --> 00:01:53,640 Of course, in this particular example, what we're interested in is a sum of these two totally separate 24 00:01:53,640 --> 00:01:56,790 random variables because we want to know the total time. 25 00:01:56,790 --> 00:02:02,190 If we add these together, the total time it takes our company to pay employees, which means we're 26 00:02:02,190 --> 00:02:07,200 going to be looking here, of course, at the sum of two random variables instead of difference. 27 00:02:07,200 --> 00:02:11,880 Now, before we go forward with actually talking about the math of these calculations, the first thing 28 00:02:11,880 --> 00:02:17,160 we want to say is that it's important that our two variables be independent of one another. 29 00:02:17,160 --> 00:02:23,670 And the reason for that is because our standard deviation value, the standard deviation of the sum 30 00:02:23,670 --> 00:02:26,970 is only going to make sense when we have independent variables. 31 00:02:26,970 --> 00:02:31,200 So these two data sets can't be affected by one another. 32 00:02:31,200 --> 00:02:37,560 But if we're confident that these two random variables X and Y are independent of each other, then 33 00:02:37,560 --> 00:02:44,160 we can find their combination and calculate a mean variance and standard deviation for that combination. 34 00:02:44,160 --> 00:02:50,340 We also want to say that it's important that our two different random variables have matching units 35 00:02:50,340 --> 00:02:52,230 or units that make sense together. 36 00:02:52,230 --> 00:02:59,130 For example, if our first random variable is given in units of ours and our second random variable 37 00:02:59,130 --> 00:03:05,820 is given in units of square meters, so we have a measure of time and a measure of area doesn't really 38 00:03:05,820 --> 00:03:09,900 make sense to add those two things together or to find their difference. 39 00:03:09,900 --> 00:03:16,560 It's nonsensical to think about adding time to some area measurement that's probably obvious, but we 40 00:03:16,560 --> 00:03:18,930 just want to make sure that our units are matching. 41 00:03:18,930 --> 00:03:24,720 So if our random variable X here is in ours, we want to make sure that our random variable Y is also 42 00:03:24,720 --> 00:03:26,550 an hour so we can get total hours. 43 00:03:26,550 --> 00:03:31,410 And of course if we were given two different data sets and maybe the first one is in terms of hours 44 00:03:31,410 --> 00:03:36,870 and the second one is in terms of minutes, we would want to convert the second data set from minutes 45 00:03:36,870 --> 00:03:42,120 to hours or the first data set from hours to minutes so that we have matching units and it makes sense 46 00:03:42,120 --> 00:03:44,940 to add the two together or to find their difference. 47 00:03:45,150 --> 00:03:51,000 But assuming that we have independence and that our units are matching, let's go ahead now and talk 48 00:03:51,000 --> 00:03:55,290 about the idea of combining these two random variables. 49 00:03:55,290 --> 00:04:01,860 Before we do that, let's look at the data sets individually, starting with the mean of each data set. 50 00:04:02,070 --> 00:04:08,790 We're going to say for now that this is the entire population of data instead of a sample, which means 51 00:04:08,790 --> 00:04:14,400 that the mean is going to be given by the sum of these hourly values here. 52 00:04:14,400 --> 00:04:18,570 One plus two is three, plus two is five, plus three is eight. 53 00:04:18,600 --> 00:04:21,839 Eight divided by the number of data points, four is two. 54 00:04:21,839 --> 00:04:26,310 So we have a mean of eight over four or two. 55 00:04:26,550 --> 00:04:31,410 And then over here we'll say this is the mean with respect to the variable X, and then over here, 56 00:04:31,410 --> 00:04:38,580 the mean of the variable Y here will be two plus three is five, plus five is ten, plus six is 16. 57 00:04:38,580 --> 00:04:42,390 So we have 16 divided by four or four. 58 00:04:42,390 --> 00:04:48,990 So what we're saying is that maybe our company has four departments in total and the manager of the 59 00:04:48,990 --> 00:04:53,800 first department takes one hour to approve timesheets, the manager, the second department takes 2 60 00:04:53,800 --> 00:04:58,410 hours, the manager of the third department takes 2 hours, and the manager of the fourth department 61 00:04:58,410 --> 00:04:59,880 takes 3 hours. 62 00:04:59,950 --> 00:05:01,390 To approve timesheets. 63 00:05:01,390 --> 00:05:08,620 And then our payroll team takes 2 hours, 3 hours, 5 hours and 6 hours to process payroll for departments 64 00:05:08,620 --> 00:05:10,950 one, two, three and four, respectively. 65 00:05:10,960 --> 00:05:18,040 So we can say that on average, managers take 2 hours to approve timesheets and on average payroll takes 66 00:05:18,040 --> 00:05:22,110 4 hours to process payroll for each department. 67 00:05:22,120 --> 00:05:24,910 So we have a mean for each random variable. 68 00:05:24,910 --> 00:05:28,180 And then of course, we could also calculate variance. 69 00:05:28,180 --> 00:05:33,820 So our variance calculation, remember, we take each data point and subtract the mean. 70 00:05:33,820 --> 00:05:36,220 So here we start with this data point of one. 71 00:05:36,220 --> 00:05:38,440 So we say one minus two. 72 00:05:38,470 --> 00:05:44,590 We square that value and then we add to that the same calculation for all of the other departments. 73 00:05:44,590 --> 00:05:54,100 So here we get two minus two quantity squared plus two minus two quantity squared plus three minus two 74 00:05:54,100 --> 00:05:55,180 quantity squared. 75 00:05:55,180 --> 00:06:03,250 And we divide this whole thing by the number of data points that we have four and we'll divide by four 76 00:06:03,250 --> 00:06:08,080 because we're going to say that this data represents the entire population of our departments. 77 00:06:08,080 --> 00:06:14,110 If this was a sample of a larger number of departments in our company, we would divide by n minus one 78 00:06:14,110 --> 00:06:16,570 or in this case four, minus one or three. 79 00:06:16,570 --> 00:06:18,880 So we'd be dividing by three instead of by four. 80 00:06:19,000 --> 00:06:23,950 But since we're going to say that this represents population, data will divide by n the total number 81 00:06:23,950 --> 00:06:24,940 of departments here. 82 00:06:25,030 --> 00:06:31,660 So the result then is one fourth, one minus two is negative, one quantity squared is a positive one. 83 00:06:31,660 --> 00:06:32,770 So we have one here. 84 00:06:32,770 --> 00:06:37,000 This will be zero zero and then three minus two is one, quantity squared is one. 85 00:06:37,240 --> 00:06:40,360 So we get two divided by four or. 86 00:06:41,150 --> 00:06:42,040 One half. 87 00:06:42,050 --> 00:06:43,970 So our variance then. 88 00:06:45,290 --> 00:06:46,760 Is one half. 89 00:06:46,910 --> 00:06:54,050 And then standard deviation is the square root of that or one over square root two. 90 00:06:54,080 --> 00:07:00,410 If we do a variance in standard deviation calculation for the variable y, we start with each data point, 91 00:07:00,410 --> 00:07:08,090 subtract the mean, so we get two minus four quantity squared plus three minus four. 92 00:07:08,950 --> 00:07:10,840 Quantity squared plus. 93 00:07:11,750 --> 00:07:14,330 Five minus four squared. 94 00:07:14,330 --> 00:07:18,890 And then finally the last data point six minus four quantity squared. 95 00:07:18,890 --> 00:07:25,310 And then again, we're doing a population calculation, assuming that this is the entire population. 96 00:07:25,310 --> 00:07:30,140 So we divide by the number of data points, which is four, or multiply that by one fourth. 97 00:07:30,140 --> 00:07:34,550 And if we simplify here, we get one fourth and then we'll get two. 98 00:07:34,550 --> 00:07:39,800 Minus four is negative two, quantity squared is positive, four, three minus four is negative, one 99 00:07:39,800 --> 00:07:47,480 quantity squared is one, five, minus four is one squared is one and six minus four is two squared 100 00:07:47,480 --> 00:07:48,620 is four. 101 00:07:48,710 --> 00:07:52,700 And so we get four plus one plus one plus four is ten. 102 00:07:52,700 --> 00:07:56,840 Ten divided by four is five halves or two and one half. 103 00:07:56,840 --> 00:08:05,870 So we can say then that the variance of Y is five halves, which means that the standard deviation of 104 00:08:05,870 --> 00:08:12,080 Y is the square root of that, or the square root of five halves, the square root of two and one half. 105 00:08:12,110 --> 00:08:17,330 Now, here's what the idea of combining random variables really comes in handy. 106 00:08:17,330 --> 00:08:22,670 Let's say that we've already been collecting this data about the amount of time that our managers spend 107 00:08:22,670 --> 00:08:28,490 approving timesheets and the amount of time that our payroll team spends processing payroll for each 108 00:08:28,490 --> 00:08:29,150 department. 109 00:08:29,420 --> 00:08:35,870 So assuming we already have all of this information, if we now want to calculate the sum or difference 110 00:08:35,870 --> 00:08:42,440 of these two data sets, we don't have to start over from this raw data and create a whole new data 111 00:08:42,440 --> 00:08:47,660 set for the sum or the difference and then go through the process again of calculating mean variance 112 00:08:47,660 --> 00:08:49,880 and standard deviation for the new dataset. 113 00:08:49,880 --> 00:08:56,180 If we already have these summary statistics, then we can go directly to these formulas and use them 114 00:08:56,180 --> 00:08:59,420 to find the sum or difference of the two variables. 115 00:08:59,420 --> 00:09:05,660 So for instance, we can see here that the mean of the sum is equal to the sum of the means. 116 00:09:05,660 --> 00:09:14,180 In other words, to find the mean of the sum, we simply add the two means together to plus four and 117 00:09:14,180 --> 00:09:20,300 we get 6 hours, which means that total payroll processing time between the managers and the payroll 118 00:09:20,300 --> 00:09:25,850 department, if we add up all that data together, our mean is 6 hours per department. 119 00:09:25,850 --> 00:09:32,420 And similarly, if we wanted to find the difference, the mean of the difference, we would simply take 120 00:09:32,420 --> 00:09:34,190 the difference of the means. 121 00:09:34,190 --> 00:09:36,500 So we could take four minus two. 122 00:09:37,670 --> 00:09:39,070 To get to. 123 00:09:39,080 --> 00:09:45,980 And we could say that on average, the payroll department spends 2 hours extra per department than the 124 00:09:45,980 --> 00:09:47,820 manager spend per department. 125 00:09:47,840 --> 00:09:51,870 On the process of getting the employees paid. 126 00:09:51,890 --> 00:09:56,360 Realize here that there are a few different ways that we can prove this to ourselves, one of which 127 00:09:56,360 --> 00:09:59,990 being that we can actually create a new data set. 128 00:09:59,990 --> 00:10:07,250 We'll call it the some data set, and we find the new data set for the some by adding data points from 129 00:10:07,250 --> 00:10:09,600 these original data sets, X and Y. 130 00:10:09,620 --> 00:10:15,800 So if we take one plus two, we get the new data point three in the data set for the sum. 131 00:10:15,800 --> 00:10:24,280 If we take two plus three, we get five, two plus five, we get seven and three plus six, we get nine. 132 00:10:24,290 --> 00:10:30,770 This is now a new data set where we've summed the data points in sets X and Y. 133 00:10:30,770 --> 00:10:36,410 And what we realize here is that if we take the mean of this new data set, we would add 3 to 5 to get 134 00:10:36,410 --> 00:10:40,160 eight plus seven is 15, plus nine is 24. 135 00:10:40,160 --> 00:10:48,230 We get 24 divided by the number of data points in the population for 24 divided by four is six and we 136 00:10:48,230 --> 00:10:50,750 see that we get back to this value. 137 00:10:50,750 --> 00:10:52,490 We already found same thing here. 138 00:10:52,490 --> 00:10:58,430 If we create a new data set for the difference, we'll say a new data set. 139 00:10:59,150 --> 00:10:59,540 Four. 140 00:10:59,690 --> 00:11:07,010 The difference if we take two minus one, we get one, three minus two, we get one five minus two, 141 00:11:07,040 --> 00:11:10,970 we get three and six minus three, we get three. 142 00:11:10,970 --> 00:11:13,130 Now we have a new data set for the difference. 143 00:11:13,130 --> 00:11:18,440 And if we take the mean here, we get one plus one is two, plus three is five, plus three is eight, 144 00:11:18,440 --> 00:11:20,660 eight divided by four is two. 145 00:11:20,660 --> 00:11:24,800 And we get back to the mean for the difference that we already found. 146 00:11:24,800 --> 00:11:30,560 Now, in the same way, if we want to make a variance calculation for the sum or the difference of X 147 00:11:30,560 --> 00:11:37,850 and Y, we don't have to build this new data set for the sum and then start fresh with a variance calculation. 148 00:11:37,850 --> 00:11:39,050 For this new data set. 149 00:11:39,050 --> 00:11:44,120 We don't have to find the data set for the difference and then start fresh with a variance calculation 150 00:11:44,120 --> 00:11:45,860 for this data set for the difference. 151 00:11:45,860 --> 00:11:52,100 Instead, we can use the fact that we already have variance values for both X and Y and use these formulas 152 00:11:52,100 --> 00:11:54,230 here to calculate variance. 153 00:11:54,230 --> 00:12:00,440 So the variance for the sum is equal to the sum of the variances. 154 00:12:00,440 --> 00:12:07,940 So we take one half plus five halves is six over two or three, so we get a variance of three. 155 00:12:07,940 --> 00:12:14,330 And then similarly here for the variance of the difference, to find the variance for the difference, 156 00:12:14,330 --> 00:12:16,460 we take the sum of the variances. 157 00:12:16,460 --> 00:12:18,470 So again we get three. 158 00:12:18,470 --> 00:12:24,440 Now while we're here, it's important to mention what we just said here, the variance formula that 159 00:12:24,440 --> 00:12:29,090 we used to find the variance for the sum and the variance for the difference was identical. 160 00:12:29,090 --> 00:12:32,810 When we find the variance for the sum, we add the variances. 161 00:12:32,810 --> 00:12:37,100 And you would think that when we find the variance for the difference, we take the difference of the 162 00:12:37,100 --> 00:12:44,510 variances like we did up here with the formulas for the mean, but instead we find the sum of the variances 163 00:12:44,510 --> 00:12:45,050 again. 164 00:12:45,050 --> 00:12:50,030 And that's because both data sets bring their own variances to the table. 165 00:12:50,030 --> 00:12:57,290 And even when we combine them into a new data set for the sum or a new data set for the difference, 166 00:12:57,290 --> 00:13:03,110 we're not eliminating any of the spread of the data set when we do that. 167 00:13:03,110 --> 00:13:09,680 And so the variance for the new data set, whether we're finding the sum or the difference as our combination, 168 00:13:09,680 --> 00:13:15,620 the variance is still going to be found by summing the two variances from the original data sets X and 169 00:13:15,620 --> 00:13:21,020 Y, which means then of course, that our standard deviations for both the sum and difference are also 170 00:13:21,020 --> 00:13:21,950 going to be the same. 171 00:13:21,980 --> 00:13:25,670 The standard deviations will be, of course, the square root of the variance. 172 00:13:25,670 --> 00:13:34,160 So the standard deviation of the sum will be square root 3 hours and the standard deviation of the difference 173 00:13:34,160 --> 00:13:36,770 will be square root 3 hours. 174 00:13:36,800 --> 00:13:44,600 Keep in mind also that it's invalid for us to find the standard deviation of the sum or difference directly. 175 00:13:44,600 --> 00:13:51,440 In other words, notice here in our table of formulas that we are combining the means and the variances, 176 00:13:51,440 --> 00:13:56,600 but we don't have formulas for the standard deviation, and that's because we can't find the standard 177 00:13:56,600 --> 00:13:59,930 deviation of the sum by summing the standard deviations. 178 00:13:59,930 --> 00:14:05,000 And we can't find the standard deviation of the difference by summing the standard deviations or taking 179 00:14:05,000 --> 00:14:06,710 the difference of the standard deviations. 180 00:14:06,710 --> 00:14:13,310 The only way to get to standard deviation for the combination is to first find the variance of the combination 181 00:14:13,310 --> 00:14:19,310 by summing the variances of the original data sets and then to take the square root of variance in order 182 00:14:19,310 --> 00:14:20,660 to get standard deviation. 183 00:14:20,660 --> 00:14:27,110 So the whole takeaway here is that if we find ourselves in a scenario where we already have summary 184 00:14:27,110 --> 00:14:32,360 statistics like these ones, we've been maybe tracking this information across two data sets, X and 185 00:14:32,360 --> 00:14:32,870 Y. 186 00:14:32,870 --> 00:14:38,900 So in this case we've been keeping track of how many hours it takes managers to approve timesheets for 187 00:14:38,900 --> 00:14:44,690 each department, and we've been keeping track of how many hours it takes our payroll team to process 188 00:14:44,690 --> 00:14:50,720 payroll for each department, such that we already have a mean variance and standard deviation value 189 00:14:50,720 --> 00:14:54,410 for both of these independent random variables X and Y. 190 00:14:54,440 --> 00:15:01,040 Then when we want to find the sum or difference of these two variables, when we want to find the combination, 191 00:15:01,040 --> 00:15:08,120 what we realize is that we do not have to start from the beginning and combine these two data sets into 192 00:15:08,120 --> 00:15:15,950 a new data set for the sum or a new data set for the difference and start completely from scratch with 193 00:15:15,950 --> 00:15:22,190 a mean variance and standard deviation calculation for this new data set or this new data set. 194 00:15:22,190 --> 00:15:28,700 Instead, if we already have this information, we can go straight to these formulas and immediately 195 00:15:28,700 --> 00:15:36,800 and directly make these calculations here to see the mean variance in standard deviation of both the 196 00:15:36,800 --> 00:15:43,370 sum of these random variables and the difference of these random variables, which just means that we 197 00:15:43,370 --> 00:15:47,900 want to keep in mind that when we already have this information, we can move quickly to the combination 198 00:15:47,900 --> 00:15:49,430 and save ourselves a lot of time. 199 00:15:49,430 --> 00:15:54,380 Redoing the math from scratch with a new combined data set. 200 00:15:54,380 --> 00:15:58,400 And of course that can be really helpful when we're trying to answer real world question. 201 00:15:58,740 --> 00:16:04,290 Like the one we were here where we were trying to get a total for the complete number of hours spent 202 00:16:04,290 --> 00:16:10,170 across our entire company collecting all departments together in order to get payroll processed from 203 00:16:10,170 --> 00:16:11,160 beginning to end. 204 00:16:11,160 --> 00:16:18,030 Or the difference between these two departments on the amount of time spent on payroll.