1 00:00:04,090 --> 00:00:11,050 In this video, we're going to look at probability density functions now, mathematics is a very precise 2 00:00:11,050 --> 00:00:16,780 and deterministic field, so a way of expressing random events and uncertainties is required. 3 00:00:18,060 --> 00:00:24,570 A random variable is a way to mathematically express steady acoustic outcomes as real numbers so we 4 00:00:24,570 --> 00:00:31,470 can write a random variable like say so X is the random variable and it has a possibility of being any 5 00:00:31,470 --> 00:00:39,030 of the real numbers in set a given all the different outcomes of an experiment in this set is so random, 6 00:00:39,030 --> 00:00:46,200 variable is simply a mapping of all the experimental outcomes or events into real numbers of that e.. 7 00:00:47,190 --> 00:00:52,560 However, once the experiment has been carried out, the value of the random variable or outcome has 8 00:00:52,560 --> 00:00:53,310 been determined. 9 00:00:53,610 --> 00:00:59,580 Once this happens, you can then treat the value of the random variable as just a normal algebraic value. 10 00:01:02,060 --> 00:01:09,020 Let's say we have a random variable X, but we like to know how likely each outcome is to occur, is 11 00:01:09,020 --> 00:01:11,250 each outcome equally likely to occur? 12 00:01:11,270 --> 00:01:17,600 So if we're talking about a crossover, TOYIN is 50/50 for either sides or some outcomes more likely 13 00:01:17,600 --> 00:01:18,800 to occur than others? 14 00:01:19,220 --> 00:01:23,170 We describe and quantify this with probability density functions. 15 00:01:24,170 --> 00:01:30,650 So probability density functions of PDF measure the relative likelihood or probability that a specific 16 00:01:30,650 --> 00:01:31,780 outcome will occur. 17 00:01:33,220 --> 00:01:39,520 So a probability density function might look like, say, on the y axis, we have the likelihood and 18 00:01:39,520 --> 00:01:44,320 on the x axis over here, we have the different values that the random variable can take. 19 00:01:44,620 --> 00:01:49,180 So in this case, the random variable can take any number of very negative Vendy and Infinity. 20 00:01:49,180 --> 00:01:51,400 It's just a real plane this way. 21 00:01:52,360 --> 00:01:56,140 And this red line here would actually be the probability density function. 22 00:01:56,140 --> 00:02:01,150 And we commonly write this as little ethics as a function of X. 23 00:02:01,810 --> 00:02:06,520 So the value of this function at each point is the likelihood of that event occurring. 24 00:02:07,550 --> 00:02:13,460 However, it is important to note that likelihood and probability are not the same thing, we can calculate 25 00:02:13,490 --> 00:02:19,520 a probability from the likelihood function, but only when we consider it from between a range of values. 26 00:02:20,210 --> 00:02:26,330 So to calculate the probability that the random variable would take a number between two values, let's 27 00:02:26,330 --> 00:02:29,710 say between A and B. We can use this equation here. 28 00:02:30,050 --> 00:02:37,910 So the probability that the random variable X is going to be between A and B is just the integral between 29 00:02:37,940 --> 00:02:40,640 A and B of the probability density function. 30 00:02:41,030 --> 00:02:45,000 So this is basically saying the area under the curve is a probability. 31 00:02:45,860 --> 00:02:51,890 So if we were to integrate the whole curve between negative infinity and infinity, by definition, 32 00:02:51,890 --> 00:02:53,660 the probability has to equal to one. 33 00:02:55,160 --> 00:03:00,980 So if we want the probability that the random variable value between zero and infinity, we have to 34 00:03:00,980 --> 00:03:06,380 calculate this area here, the area on the curve for the positive x axis. 35 00:03:08,040 --> 00:03:13,140 So let's have a look at a more concrete example, let's say we have a probability density function that 36 00:03:13,140 --> 00:03:18,360 looks like this, so we have an even probability that we'll get a random variable number between negative 37 00:03:18,360 --> 00:03:19,380 one and one. 38 00:03:20,160 --> 00:03:22,980 So we're basically saying the likelihood value is just a constant value. 39 00:03:23,010 --> 00:03:29,610 So if X is just going to equal 2.5 between negative one and one and everywhere else, it's going to 40 00:03:29,610 --> 00:03:30,240 be zero. 41 00:03:32,020 --> 00:03:37,390 So let's say we want to know the probability that the random variable X is going to take on a value 42 00:03:37,390 --> 00:03:39,620 of negative point seventy five. 43 00:03:39,940 --> 00:03:45,210 The answer is actually just going to be zero because we want to look at the area under the curve. 44 00:03:45,220 --> 00:03:50,710 So if we just try to evaluate the likelihood function at a single point such as this point here, it 45 00:03:50,710 --> 00:03:56,140 has zero area because they are only looking at a single point to calculate the probability we actually 46 00:03:56,140 --> 00:03:57,220 have to take the integral. 47 00:03:57,670 --> 00:04:03,520 So now let's say we want to know the probability that the random variable will be between negative one 48 00:04:03,700 --> 00:04:05,690 and negative point seventy five. 49 00:04:05,980 --> 00:04:11,330 Well, then we can just calculate the area of the square here so we know the value is point five here. 50 00:04:11,590 --> 00:04:14,290 We know the distance is O point to five. 51 00:04:14,610 --> 00:04:18,160 So therefore the area is just zero point one two five. 52 00:04:18,190 --> 00:04:23,650 So the probability that the random variable X takes a value between negative one and negative point 53 00:04:23,650 --> 00:04:26,020 seventy five is just this area here. 54 00:04:26,410 --> 00:04:29,920 So this is three point twenty five multiplied by point five. 55 00:04:30,430 --> 00:04:32,620 So we get three point one, two five. 56 00:04:34,040 --> 00:04:39,200 So this is how we can use the probability density functions to work out probability, so the function 57 00:04:39,200 --> 00:04:42,720 graphs likelihood and the area under the graph is probability. 58 00:04:43,130 --> 00:04:49,310 Now, the probability of the whole area has to equal one, because we know that out of all the cases, 59 00:04:49,310 --> 00:04:52,310 the random variable has to take on a single value. 60 00:04:54,140 --> 00:04:59,840 Now that we have defined what a payday for a random variable is and how we can calculate the probability 61 00:04:59,840 --> 00:05:06,580 of certain events happening, a cooperation that can be applied to any PDF is the expectation operator 62 00:05:06,860 --> 00:05:08,870 and that's given by this equation here. 63 00:05:09,260 --> 00:05:11,690 So basically the expectation operator. 64 00:05:11,690 --> 00:05:15,510 So E of X is just the integral of the whole density function. 65 00:05:15,530 --> 00:05:21,680 So from negative Phinney to infinity of the random variable X times the probability density function. 66 00:05:22,640 --> 00:05:28,310 To give the expectation operator a bit more meaning, this is basically calculating the main of the 67 00:05:28,310 --> 00:05:29,770 probability density functions. 68 00:05:29,780 --> 00:05:36,650 So the expected value or main of the random variable is usually for simplicity, just as this X bar 69 00:05:36,860 --> 00:05:42,890 or little expo is just the expected value of the random variable probability density function. 70 00:05:44,510 --> 00:05:49,500 So the expectation over it allows us to investigate some distribution statistical properties. 71 00:05:50,210 --> 00:05:55,700 It is very useful to be able to describe a probability density function or distribution in a few key 72 00:05:55,700 --> 00:05:56,380 properties. 73 00:05:56,630 --> 00:06:02,510 And the first one we've already covered, which is the main and some other common parameters, are the 74 00:06:03,080 --> 00:06:04,940 variance and the skew. 75 00:06:05,870 --> 00:06:11,240 And each of these distribution statistics can be calculated using the expectation operator. 76 00:06:13,330 --> 00:06:18,490 So first, let's have a look at the main, so let's say we have a probability density function that 77 00:06:18,490 --> 00:06:19,250 looks like this. 78 00:06:19,960 --> 00:06:23,660 Now, the main is just going to be the average of the distribution. 79 00:06:23,680 --> 00:06:30,010 So in this case, it's going to be at this point here and we can say that this is X, so this is the 80 00:06:30,010 --> 00:06:37,090 main or the random variable distribution and simply just calculated from the expectation operator of 81 00:06:37,090 --> 00:06:38,020 the random variable. 82 00:06:40,130 --> 00:06:45,800 The distribution variance looks at how spread the distribution is, so you can see on this distribution 83 00:06:45,800 --> 00:06:50,960 here, we can say it's got quite a large variance, is quite large compared to a distribution that has 84 00:06:50,960 --> 00:06:53,330 a smaller variance, is more tightly compact. 85 00:06:53,980 --> 00:06:59,760 So we call this parameter the variance and it's basically a measure of the spread of the distribution. 86 00:06:59,990 --> 00:07:06,710 So this is an Sigma X, so the variance of a distribution is a measure of how much the distribution 87 00:07:06,710 --> 00:07:12,460 varies away from the main value here and again is calculated using the expectation of radar. 88 00:07:12,860 --> 00:07:18,380 So using the expectation operator, we calculate the random variable distribution away from the main 89 00:07:18,380 --> 00:07:19,220 and we square it. 90 00:07:19,520 --> 00:07:24,800 And this is going to give us the variance or the sigma squared, the standard deviation squared. 91 00:07:26,000 --> 00:07:31,070 We can also expand this top equation and write the variance in a slightly different form we get the 92 00:07:31,070 --> 00:07:35,570 variance is just going to be the expected value squared minus the main squared. 93 00:07:36,700 --> 00:07:42,610 The main and the variance give us quite a bit of information about the distribution and effect is going 94 00:07:42,610 --> 00:07:47,230 to be the two main parameters when we're looking at the Gaussian or normal distribution. 95 00:07:48,310 --> 00:07:51,790 So we could actually write the distribution to be random variable. 96 00:07:51,790 --> 00:07:57,730 X is a distribution that has a main export and variance sigma squared X. 97 00:07:58,820 --> 00:08:04,580 And then lastly, we can look at the distribution skewness, so the skewness of the distribution looks 98 00:08:04,580 --> 00:08:06,050 about how symmetrical is. 99 00:08:06,080 --> 00:08:12,290 So in this case, we have a very symmetrical distribution compared to this case where we have a non-special 100 00:08:12,380 --> 00:08:15,410 distribution, we have a very long tail on one side. 101 00:08:15,650 --> 00:08:20,110 So this distribution is going to be more skewed towards one way than the other. 102 00:08:20,780 --> 00:08:27,350 And we can calculate the Skewness by basically looking at the expectation of radar for the random variable 103 00:08:27,680 --> 00:08:33,740 difference away from the main but raised to the power of three instead of the power to for the variance 104 00:08:34,920 --> 00:08:37,760 so we can calculate the skew as the expectation of rate. 105 00:08:38,300 --> 00:08:44,300 But we can also look at the coefficient of skew or the skewness, just the value of skew divided by 106 00:08:44,300 --> 00:08:46,640 the standard deviation to the power of three. 107 00:08:48,290 --> 00:08:54,380 So the three main statistical properties of a distribution can be the main, which is just the expectation 108 00:08:54,380 --> 00:08:56,960 of radar, which is also commonly called the first moment. 109 00:08:57,560 --> 00:09:02,470 The variance or second moment is just a difference away from the main square. 110 00:09:02,900 --> 00:09:07,790 And then the skew is the third element, which is just the difference away from the main to the power 111 00:09:07,790 --> 00:09:08,270 of three. 112 00:09:09,930 --> 00:09:15,750 In this video, we have quickly covid what random variables are, how we can describe the distribution 113 00:09:15,750 --> 00:09:20,900 or random variables, so how we can describe how likely certain values of the variable are. 114 00:09:21,270 --> 00:09:25,500 And then we looked at statistical properties to describe these distributions.