1 00:00:04,000 --> 00:00:09,490 Random variables can be transformed through a function, suppose that we have a random variable X and 2 00:00:09,490 --> 00:00:10,840 it's associated PDF. 3 00:00:11,200 --> 00:00:17,980 It is possible to apply a mathematical function to the BBF to find the transform PDF for the transform 4 00:00:17,980 --> 00:00:18,740 random variable. 5 00:00:19,600 --> 00:00:27,160 So if we have a random variable X and we have the associated PDF, if X, it is possible to apply a 6 00:00:27,160 --> 00:00:35,530 mathematical function so Y easily equals G of X to the PDF to find the PDF of the transform variable 7 00:00:35,680 --> 00:00:37,840 Y, which is the F. 8 00:00:37,840 --> 00:00:38,260 Why. 9 00:00:39,720 --> 00:00:45,880 So let's assume the transfer functions are monotonic so they can be monotonically increasing or decreasing. 10 00:00:45,900 --> 00:00:51,620 It doesn't matter as long as this transformation function, this why of G of X is monotonic. 11 00:00:52,170 --> 00:00:59,130 So we have the transformation function, why they call G of X. Now we want to find the inverse mapping. 12 00:00:59,130 --> 00:01:05,340 So we want to find the function, the inverse G that maps from the Y variable to back to the X variable. 13 00:01:05,730 --> 00:01:10,560 And we're going to rename this inverse mapping the inverse G into H. 14 00:01:11,840 --> 00:01:17,690 We then can express the transformation of the PDF using this function here, so we won't go through 15 00:01:17,690 --> 00:01:19,100 the derivation of this function. 16 00:01:19,130 --> 00:01:25,190 You can look at the associated notes if you want to go into more detail, but this is the function that 17 00:01:25,190 --> 00:01:33,320 will transform the PDF effects into the PDF if y so this takes the random variable X and works out the 18 00:01:33,320 --> 00:01:36,620 random variable Y and the had pdf of it. 19 00:01:37,040 --> 00:01:42,200 So you can say it's pretty much the derivative of the inverse function, the absolute value of it. 20 00:01:42,680 --> 00:01:48,590 Thom's pi the PDF for the X variable with the function substituted into it. 21 00:01:51,320 --> 00:01:56,420 So using this relationship here, we then can look into the example of a linear transformation of a 22 00:01:56,420 --> 00:01:58,780 Gaussian PDF and calculate the result. 23 00:02:00,290 --> 00:02:05,840 So let's suppose we have a random variable X and we know that it is the normal distribution or Gaussian 24 00:02:05,840 --> 00:02:11,070 distribution and it has a mean of X bar and a variant of Sigma X squared. 25 00:02:11,870 --> 00:02:18,850 We know from the previous video that this here is a function of the PDF for this normal distribution. 26 00:02:19,370 --> 00:02:22,190 So we have the name here and the variance here. 27 00:02:23,300 --> 00:02:27,500 So now suppose that we have the transformation and we want to do a linear transformation. 28 00:02:27,500 --> 00:02:31,380 So we have A X plus B equals Y. 29 00:02:31,430 --> 00:02:36,380 So this is going to be a linear transformation function that we're going to apply to this random variable. 30 00:02:37,130 --> 00:02:44,270 And in this of course, this coefficient A and B have to be real numbers and A cannot be equal to zero. 31 00:02:44,300 --> 00:02:46,080 Otherwise, it's not a transformation. 32 00:02:47,000 --> 00:02:51,350 So the first thing we want to do is to work out the inverse G mapping function. 33 00:02:51,860 --> 00:02:56,450 So that's just going to be Y minus B divided by A.. 34 00:02:56,480 --> 00:03:00,940 So this is going to be our investment in function and we're going to define it to be hech. 35 00:03:02,980 --> 00:03:07,300 Now we know that we need to get the derivative of its derivative have to exist to be able to do this 36 00:03:07,300 --> 00:03:13,750 mapping and the derivative of the function is simply just one over a that's fairly straightforward from 37 00:03:13,750 --> 00:03:15,880 first order differentiation. 38 00:03:17,470 --> 00:03:22,630 So using the relationship that we presented on the previous slide, we can now fill in all this other 39 00:03:22,630 --> 00:03:23,310 information. 40 00:03:23,890 --> 00:03:29,620 So first we know that the derivative is just one everyday, so we can fill that in here. 41 00:03:31,170 --> 00:03:37,370 Next, we want to work out the second time here, so the PDF for X with the function substituted in. 42 00:03:37,650 --> 00:03:44,180 So what we can do is we can take our Hej function substituted in for X into the Expedia. 43 00:03:45,300 --> 00:03:51,540 Once we've done that substitution, we can then take the whole function and fill it in for this term 44 00:03:51,540 --> 00:03:51,820 here. 45 00:03:52,470 --> 00:03:55,830 So if we do all this, we end up with this equation here. 46 00:03:55,860 --> 00:04:03,210 So this is the transformation and this gives us the PDF for the Y random variable after we do the transformation. 47 00:04:03,900 --> 00:04:08,430 So you can see that this function here looks very similar to this function over here. 48 00:04:09,000 --> 00:04:12,180 So in fact, they're both Gaussian distributions. 49 00:04:12,510 --> 00:04:18,810 The only difference you can see is that we've changed the meaning of the Gaussian distribution and we've 50 00:04:18,810 --> 00:04:21,560 also changed the variance of the Gaussian distribution. 51 00:04:22,470 --> 00:04:28,410 So equating the two parameters, we can work out the new meaning of this random variable y it's just 52 00:04:28,410 --> 00:04:34,490 a time to previous mean plus B, so this is basically just doing the transformation on the main. 53 00:04:34,530 --> 00:04:35,810 So it's fairly straightforward. 54 00:04:36,660 --> 00:04:41,800 Also, we can see the variance here is just being scaled by the scale factor, eh. 55 00:04:42,090 --> 00:04:47,460 So the new variance for the Y pdf is just going to be a squared time. 56 00:04:47,470 --> 00:04:48,210 The variance. 57 00:04:51,080 --> 00:04:56,630 So we can see that the linear transformation of a Gaussian PDF is just another Gaussian PDF with the 58 00:04:56,630 --> 00:04:58,520 mean and variance transformed. 59 00:04:59,030 --> 00:05:07,190 So here's the original PDF of of main exposure and variance Sigma X, we apply the mapping function 60 00:05:07,190 --> 00:05:14,000 of A, X plus B, so a linear transformation to get the new wife the random variable. 61 00:05:14,610 --> 00:05:20,600 All we have to do is we just have to transform the main using this equation here and transform the variance 62 00:05:20,930 --> 00:05:22,220 using this equation here. 63 00:05:23,600 --> 00:05:29,540 So this is a major take for a Bayesian probabilistic estimation, a linear transformation of a Gaussian 64 00:05:29,540 --> 00:05:31,700 PDF is just another Gaussian PDF. 65 00:05:32,300 --> 00:05:38,090 If we can express the PDF as a Gaussian, then we can dramatically simplify the mathematics involved 66 00:05:38,090 --> 00:05:43,700 in estimation when we're working with probabilities in particular, we won't we will not have to carry 67 00:05:43,700 --> 00:05:50,290 out integrations or payday's to a probability which may or may not have close form solutions. 68 00:05:50,570 --> 00:05:57,400 So this is why when we're doing probabilistic data fusion, that ideally we'd like to use Gaussian distributions 69 00:05:57,410 --> 00:06:01,220 because we can apply these simple transformations and find the results.