1 00:00:00,510 --> 00:00:06,660 Let's say we have a random variable X, but we would like to know how likely each outcome is to occur, 2 00:00:07,140 --> 00:00:11,650 is each outcome equally as likely to occur like a toss of a coin? 3 00:00:11,670 --> 00:00:17,050 We got the heads or tails will be 50 50 or some outcomes more likely to occur than others. 4 00:00:17,730 --> 00:00:20,060 We want a way to describe and quantify this. 5 00:00:20,490 --> 00:00:23,010 This is done through probability density functions. 6 00:00:24,630 --> 00:00:30,420 Probability density functions can be classified into two different types based on whether they are continuous 7 00:00:30,420 --> 00:00:31,110 or discrete. 8 00:00:31,830 --> 00:00:36,720 So probability density functions or PD are used for continuous random variables. 9 00:00:37,200 --> 00:00:42,570 Probability mass functions or PMS are used for discrete random variables. 10 00:00:42,960 --> 00:00:49,230 And each of these functions is a measure of the relative likelihood or probability that a specific outcome 11 00:00:49,230 --> 00:00:49,790 will occur. 12 00:00:51,860 --> 00:00:58,130 So let's look at the probability mass function for the outcome of rolling the dice so the outcomes of 13 00:00:58,130 --> 00:00:59,230 the dice are discrete. 14 00:00:59,240 --> 00:01:06,020 So we're going to use a PMF rather than a PDF, and we write the probability mass function as is to 15 00:01:06,020 --> 00:01:06,550 map here. 16 00:01:06,560 --> 00:01:14,390 So we have the little P capital X, explain the random variable representing the dice roll and being 17 00:01:14,390 --> 00:01:15,530 the discrete values. 18 00:01:17,760 --> 00:01:24,810 So we know from basic probability that the likelihood of the random variable being one being a single 19 00:01:24,810 --> 00:01:27,070 dot on the face is one of the six. 20 00:01:28,140 --> 00:01:32,950 We also know that the likelihood of it happening to is another one over six. 21 00:01:33,270 --> 00:01:37,680 And we can do this for all the different outcomes of the experiment, for the rolling the dice. 22 00:01:37,800 --> 00:01:42,540 So what we actually end up here, if we look at the probability mass function, we can see that it's 23 00:01:42,540 --> 00:01:43,530 a constant function. 24 00:01:44,310 --> 00:01:51,390 Each of these values here are equally as likely as each other and the probability is just one over six. 25 00:01:53,420 --> 00:02:00,260 So we can say that the probability mass function is just going to be the probability that the random 26 00:02:00,380 --> 00:02:02,870 variable equals that particular number. 27 00:02:03,290 --> 00:02:06,440 So in this case, of course, the whole function is just going to be constant. 28 00:02:06,440 --> 00:02:07,190 One of the six. 29 00:02:09,770 --> 00:02:16,310 Now, if we look at another example, what happens if we had a weighted dice, so each of the outcomes 30 00:02:16,310 --> 00:02:18,170 not as equally likely. 31 00:02:18,350 --> 00:02:26,120 So in this case, instead of having this PMF, we're going to say, well, the six face is more weighted. 32 00:02:26,120 --> 00:02:27,000 It's more likely. 33 00:02:27,020 --> 00:02:33,740 So we're going to have a likelihood of rolling a six point seventy five and the rest of the cases are 34 00:02:33,740 --> 00:02:36,860 going to have this lower probability of zero point zero five. 35 00:02:37,610 --> 00:02:42,500 So we can see from this probability mass function here that there's a lot more likelihood of rolling 36 00:02:42,500 --> 00:02:44,500 a six than any of the other ones. 37 00:02:46,040 --> 00:02:52,600 So we know from basic probability that some of all the different possibilities has to equal one. 38 00:02:52,610 --> 00:02:54,360 And we can easily see that from here. 39 00:02:54,380 --> 00:03:01,120 So if we have five times zero point five plus our point seventy five equals one. 40 00:03:01,130 --> 00:03:03,700 So it just follows the basic laws of probability as well. 41 00:03:05,560 --> 00:03:12,160 We might want to ask ourselves another question, what is the probability of rolling a fire, a number 42 00:03:12,160 --> 00:03:13,510 between one and five? 43 00:03:13,930 --> 00:03:19,780 Well, we can just use our basic probability laws and we can just sum up the probability of each of 44 00:03:19,780 --> 00:03:20,460 these events. 45 00:03:20,890 --> 00:03:25,510 So if we do that, we end up with point to five as expected. 46 00:03:26,020 --> 00:03:30,300 And of course, like I was saying before, it still follows the basic probability laws. 47 00:03:30,310 --> 00:03:35,950 So another way to come up with this number here is to say, well, we know the probability has to equal 48 00:03:35,950 --> 00:03:39,990 one and we know the probability of six is point seven five. 49 00:03:40,390 --> 00:03:44,860 Therefore, the probability of knowing anything other than a six has to be one minus the probability 50 00:03:44,860 --> 00:03:45,700 of rolling a six. 51 00:03:45,940 --> 00:03:51,250 And again, that's just give me one minus point seven five, which is point to five. 52 00:03:53,300 --> 00:03:58,310 So on the left here, we have the probability mass function of PMF that we just looked at and this is 53 00:03:58,310 --> 00:04:03,060 for this great random variables and we use the P of X to denote it. 54 00:04:03,740 --> 00:04:08,510 Now, when we look at continuous random variables, we have to use a probability density function or 55 00:04:08,510 --> 00:04:09,200 PDF. 56 00:04:09,620 --> 00:04:16,040 And the notation now changes to this F of X, so a small F capital X X for the random variable. 57 00:04:17,120 --> 00:04:22,530 So we've shown an example, probability distribution here and this is actually called a uniform distribution. 58 00:04:22,970 --> 00:04:27,890 This just means that the random variable can take on any number between two different limits and each 59 00:04:27,890 --> 00:04:29,390 number is equally as likely. 60 00:04:29,750 --> 00:04:34,790 So this is basically saying that when we query this random variable X, we're going to get a random 61 00:04:34,790 --> 00:04:39,680 number between negative one and one, and it could be any number in between these two limits. 62 00:04:41,640 --> 00:04:48,510 Now, if the probability mass function, the actual value of the function at any of the given outcomes 63 00:04:48,510 --> 00:04:50,640 is actually just the probability of the event. 64 00:04:50,970 --> 00:04:56,760 So if you want to know what the probability of beginning a four here, we just read the value and is 65 00:04:56,760 --> 00:04:58,500 going to be the probability of that event. 66 00:05:01,370 --> 00:05:06,680 We also know that when we sum up all the different probabilities for the probability mass function, 67 00:05:06,680 --> 00:05:07,700 it has to equal one. 68 00:05:07,930 --> 00:05:09,970 This is just the basic laws of probability. 69 00:05:10,790 --> 00:05:14,720 We have an equivalent relationship for the continuous function. 70 00:05:15,020 --> 00:05:20,840 So if we basically integrate the probability density function between negative Phinney and Infinity, 71 00:05:21,260 --> 00:05:25,370 the area under the curve of the probability density function has to equal one. 72 00:05:26,600 --> 00:05:32,750 So basically, if we look at this uniform distribution here, if we know that the area between Nagler 73 00:05:32,750 --> 00:05:38,950 one and one has to equal one in total, then this value here has to be point five. 74 00:05:38,990 --> 00:05:46,460 This is just simple math, because the area here, yeah, this is the area like two by a three point 75 00:05:46,460 --> 00:05:47,690 five equals one. 76 00:05:49,230 --> 00:05:54,660 So if we want to work out the probability of getting a certain number, so let's say we want to work 77 00:05:54,660 --> 00:05:59,320 out the value, the probability of getting negative point, turn five here. 78 00:05:59,880 --> 00:06:06,610 Now, if we look at the value here, if we evaluate the PDF for this value, we get a value of five. 79 00:06:06,680 --> 00:06:09,050 Now, this can't be the probability. 80 00:06:09,270 --> 00:06:12,330 The probability point five means there's a 50/50 chance of getting it. 81 00:06:12,840 --> 00:06:15,950 So this function here does not actually give us a probability. 82 00:06:16,470 --> 00:06:21,060 And the reason for this can be actually quite simple if we think about it, because over here in the 83 00:06:21,060 --> 00:06:24,590 discrete domain, we have options one to six on this example. 84 00:06:24,600 --> 00:06:29,310 So the probability of getting any one of these events or options is just one of the six. 85 00:06:29,970 --> 00:06:34,860 However, in the continuous domain, we have infinite amount of options between negative one and one. 86 00:06:35,190 --> 00:06:36,550 This is because it's continuous. 87 00:06:37,050 --> 00:06:42,660 So if we look at looking at basic probability, we would just be having we'll just be trying to get 88 00:06:42,660 --> 00:06:43,860 one over infinity. 89 00:06:44,010 --> 00:06:46,060 And this is actually equal to zero. 90 00:06:46,650 --> 00:06:51,060 So this is basically saying the probability of getting the random variable to be equal to a specific 91 00:06:51,060 --> 00:06:54,510 number in this case, negative point seven five zero. 92 00:06:55,710 --> 00:07:01,920 So this value that dysfunction outputs is not the probability now to actually get the probability out 93 00:07:01,920 --> 00:07:02,250 of this. 94 00:07:02,250 --> 00:07:06,150 We actually have to look at the integration between two different limits. 95 00:07:06,630 --> 00:07:12,300 So what this probability density function that you're giving us is giving us of the likelihood and to 96 00:07:12,300 --> 00:07:18,330 convert the likelihood into the probability we have to integrate it between two limits because it's 97 00:07:18,330 --> 00:07:19,560 easy to see in this case. 98 00:07:19,560 --> 00:07:23,280 If we integrate it between negative one and one, we're going to get one. 99 00:07:23,730 --> 00:07:27,330 If we integrate at between zero and one, we're going to get five. 100 00:07:27,540 --> 00:07:34,920 And that's saying there's a 50/50 chance of getting a number between zero and one or between zero and 101 00:07:34,920 --> 00:07:35,490 negative one. 102 00:07:36,720 --> 00:07:41,730 So this is the main difference between the probability mass function and the probability density function. 103 00:07:42,030 --> 00:07:46,980 In this great version, we can read off the probability directly from the function itself, but in the 104 00:07:46,980 --> 00:07:53,490 continuous domain, since this infinite amount of options between each number on the real domain, we 105 00:07:53,490 --> 00:07:56,430 have to look at the area under the curve to get the probability.