1 00:00:01,699 --> 00:00:19,600 Hello, welcome to another module in this massive open online course on probability and random 2 00:00:19,600 --> 00:00:21,280 variables for wireless communications. 3 00:00:21,280 --> 00:00:27,020 Today, we are going to start with a new topic that is the concept of a continuous random 4 00:00:27,020 --> 00:00:28,020 variables. 5 00:00:28,020 --> 00:00:41,720 So we are going to start dealing with the notion of a random variable which is an important 6 00:00:41,720 --> 00:00:42,720 concept. 7 00:00:42,720 --> 00:00:53,290 So we will start dealing with a random variable and a random variable can take basically as 8 00:00:53,290 --> 00:00:57,770 the name implies, can take values randomly. 9 00:00:57,770 --> 00:01:01,470 So a random variable, X. 10 00:01:01,470 --> 00:01:21,149 If X is a random variable, then X can take values randomly from either the entire set 11 00:01:21,149 --> 00:01:24,780 of real numbers or a subset of real numbers. 12 00:01:24,780 --> 00:02:02,590 So X which is a random variable can take values randomly from entire or subset of real numbers. 13 00:02:02,590 --> 00:02:05,869 So X is a continuous random variable. 14 00:02:05,869 --> 00:02:11,090 It can take values randomly from the set of real numbers. 15 00:02:11,090 --> 00:02:16,810 And this random variable X is characterized by a very important function. 16 00:02:16,810 --> 00:02:20,390 This is known as the probability density function. 17 00:02:20,390 --> 00:02:29,209 That is, this is characterized by F of X, the subscript X denotes the random variable. 18 00:02:29,209 --> 00:02:37,569 The small X denotes the value of this probability density function at a particular point X. 19 00:02:37,569 --> 00:02:51,799 So this is the probability density function also 20 00:02:51,799 --> 00:02:54,799 denoted by PDF. 21 00:02:54,799 --> 00:03:04,650 This is the probability density function or PDF of the random variable X, PDF of the random 22 00:03:04,650 --> 00:03:15,650 variable 23 00:03:15,650 --> 00:03:16,650 X. 24 00:03:16,650 --> 00:03:22,439 And what does this PDF, probability density function of the random variable X, what does 25 00:03:22,439 --> 00:03:23,439 it to denote? 26 00:03:23,439 --> 00:03:27,620 What does the probability density function of this random variable, X denote? 27 00:03:27,620 --> 00:03:33,690 Let us consider a small interval of width dX around X. 28 00:03:33,690 --> 00:03:39,439 That is let us consider a small interval between X and X plus dx. 29 00:03:39,439 --> 00:03:47,830 So let us consider a small interval between X and let us say this is the PDF. 30 00:03:47,830 --> 00:03:57,510 Let us consider a small interval, infinitesimally small interval between X and X plus dX. 31 00:03:57,510 --> 00:04:04,250 So this is my point X, this is my point X plus dX. 32 00:04:04,250 --> 00:04:10,329 We are considering an infinitesimally small interval of width dX. 33 00:04:10,329 --> 00:04:16,639 This is my probability density function F of X of X. 34 00:04:16,639 --> 00:04:45,290 So we are considering, so consider an infinitesimal interval X, X plus dX. 35 00:04:45,290 --> 00:04:49,980 That is we are considering an infinitesimal interval of width dX around X. 36 00:04:49,980 --> 00:04:54,280 That is, the interval between X and X plus dX. 37 00:04:54,280 --> 00:05:08,310 Then the quantity, F of X times dX, then if we look at this quantity, F of X times dX, 38 00:05:08,310 --> 00:05:32,520 this represents, this is the probability that random variable 39 00:05:32,520 --> 00:05:44,610 X lies in the interval X and to X plus dX. 40 00:05:44,610 --> 00:05:51,530 X lies in the interval X to X plus dX. 41 00:05:51,530 --> 00:06:05,820 So if we look at this infinitesimal interval between X and X plus dX, the quantity F of 42 00:06:05,820 --> 00:06:17,770 X times X dX, so the quantity, so the probability density function is F of X of X. Correct? 43 00:06:17,770 --> 00:06:26,490 And F of X of X times dX basically at a point X denotes the probability that the random 44 00:06:26,490 --> 00:06:34,070 variable, capital X takes value in this infinitesimal interval of width dX. 45 00:06:34,070 --> 00:06:41,040 That is, in this infinitesimal interval around X which is the infinitesimal interval X to 46 00:06:41,040 --> 00:06:42,040 X plus dX. 47 00:06:42,040 --> 00:06:50,720 so F of X of X times dX represents the probability that the random variable takes a value in 48 00:06:50,720 --> 00:06:53,090 this infinitesimal interval. 49 00:06:53,090 --> 00:06:56,620 So that is the significance of this probability density function. 50 00:06:56,620 --> 00:07:00,170 This is the significance of this probability density function. 51 00:07:00,170 --> 00:07:08,740 It denotes the probability but it is not straightforward that is F of X of X is not the probability, 52 00:07:08,740 --> 00:07:16,290 F of X of X times dX, that denotes the probability that it takes a value in the infinitesimal 53 00:07:16,290 --> 00:07:19,220 interval of width dX around X. 54 00:07:19,220 --> 00:07:26,000 And naturally, since F of X of X represents the PDF, represents the probability, it must 55 00:07:26,000 --> 00:07:33,490 be the case that since probability cannot be 0, F of X of X has to be greater than or 56 00:07:33,490 --> 00:07:34,490 equal to 0. 57 00:07:34,490 --> 00:07:42,300 That is, the PDF, the probability density function, PDF stands for the probability density 58 00:07:42,300 --> 00:07:52,220 function, is greater than or equal to 0 for all values of X. 59 00:07:52,220 --> 00:07:58,550 At any point X, the probability density function has to always be greater than or equal to 60 00:07:58,550 --> 00:07:59,550 0. 61 00:07:59,550 --> 00:08:03,900 It cannot happen that the probability density function is negative for a certain value of 62 00:08:03,900 --> 00:08:04,900 X. Okay? 63 00:08:04,900 --> 00:08:10,270 So the probability density function represents the probability density of the random variable 64 00:08:10,270 --> 00:08:17,890 X which when multiplied by dX gives the probability that the random variable X lies in the infinitesimal 65 00:08:17,890 --> 00:08:24,090 interval around small X that is lies in the infinitesimal interval between X and X plus 66 00:08:24,090 --> 00:08:25,090 dX. 67 00:08:25,090 --> 00:08:34,710 Also, this naturally means that, therefore now the probability that X lies in this interval, 68 00:08:34,710 --> 00:09:07,210 if we consider any interval, AB, probability that X lies 69 00:09:07,210 --> 00:09:18,510 in this interval AB, is basically now if you look at the probability density function again, 70 00:09:18,510 --> 00:09:24,570 this is the probability density function, F of X of X. 71 00:09:24,570 --> 00:09:32,240 And now let us say, I am considering this interval between 2 points, A and B. The probability 72 00:09:32,240 --> 00:09:40,010 that this random variable lies in the interval A to B is the area under the probability density 73 00:09:40,010 --> 00:09:52,500 function between the points A and B that is equal to integral A to B of F of X of X dX. 74 00:09:52,500 --> 00:10:00,810 Therefore what we are saying is that the probability that the random variable X lies in this interval 75 00:10:00,810 --> 00:10:07,320 between A and B is the area under the probability density function between these 2 points A 76 00:10:07,320 --> 00:10:08,529 and B. 77 00:10:08,529 --> 00:10:15,490 Therefore the probability, I can talk about the probability of X, the random variable 78 00:10:15,490 --> 00:10:20,000 X which is a continuous random variables that is it takes values on the real line. 79 00:10:20,000 --> 00:10:28,990 The probability that X is an element of A times B is equal to integral of A to B, F 80 00:10:28,990 --> 00:10:41,240 of X of X dX which is the same thing as saying that the probability X element of AB is basically 81 00:10:41,240 --> 00:10:49,560 this quantity if you can look at this, this quantity A to B of F of X dX is basically 82 00:10:49,560 --> 00:10:56,940 nothing but the area under the probability density function. 83 00:10:56,940 --> 00:11:08,540 This quantity is basically the area under the PDF that is the probability density function 84 00:11:08,540 --> 00:11:13,630 between the points A, B. 85 00:11:13,630 --> 00:11:20,590 So if I look at the interval A, B, the area under the probability density function between 86 00:11:20,590 --> 00:11:26,779 these points, A and B that is integral, A to B of F of X dX represents the probability 87 00:11:26,779 --> 00:11:34,180 that the random variable X is an element or belongs to this interval between A and B or 88 00:11:34,180 --> 00:11:42,340 that the random variable X takes values from this interval A to B. 89 00:11:42,340 --> 00:11:50,900 Naturally therefore, it follows that I am saying naturally because it follows from the 90 00:11:50,900 --> 00:12:06,750 above result, if we consider integral minus infinity to infinity, F of X of X dX, that 91 00:12:06,750 --> 00:12:09,470 must be equal to 1. 92 00:12:09,470 --> 00:12:15,750 If I consider the integral between the limits minus infinity to infinity of F of X dX where 93 00:12:15,750 --> 00:12:20,680 F of X is the probability density function, this integral over the entire set of real 94 00:12:20,680 --> 00:12:25,840 numbers must be infinity because the entire set of real numbers is the sample space. 95 00:12:25,840 --> 00:12:31,000 It is the sample space for this random variable, therefore the probability that it takes any 96 00:12:31,000 --> 00:12:34,270 value from the sample space must be 1. 97 00:12:34,270 --> 00:12:39,420 The total probability that it takes any value from the Sample space must be equal to 1. 98 00:12:39,420 --> 00:12:47,900 So what we are saying is, for this random variable, X, the sample space equals minus 99 00:12:47,900 --> 00:12:55,420 infinity to infinity, therefore probability so what we are asking is this is basically 100 00:12:55,420 --> 00:13:06,750 integral over minus infinity to infinity F of X of X dX is basically the probability 101 00:13:06,750 --> 00:13:14,870 that the random variable X belongs to minus infinity to infinity which is the same as 102 00:13:14,870 --> 00:13:22,120 the probability that the random variable, X belongs to the sample space which is equal 103 00:13:22,120 --> 00:13:23,339 to 1. 104 00:13:23,339 --> 00:13:27,839 Because this is basically, this quantity as we had seen in the axioms of probability before, 105 00:13:27,839 --> 00:13:35,930 this quantity is basically the probability of the sample space. 106 00:13:35,930 --> 00:13:39,900 And from the exempts of probability, that is from the 2nd axiom of probability, we have 107 00:13:39,900 --> 00:13:42,580 the probability of the sample space equal to 1. 108 00:13:42,580 --> 00:13:48,060 Therefore the total area under the probability density so we have 2 interesting properties, 109 00:13:48,060 --> 00:13:54,350 one is that the probability density function F of X of X is always greater than or equal 110 00:13:54,350 --> 00:13:55,630 to 0. 111 00:13:55,630 --> 00:14:02,190 And 2, the area, the integral between minus infinity to infinity F of X of dX and what 112 00:14:02,190 --> 00:14:03,190 is this? 113 00:14:03,190 --> 00:14:08,220 This is the total area under the probability density function on the entire real line between 114 00:14:08,220 --> 00:14:10,530 the limits, minus infinity to infinity. 115 00:14:10,530 --> 00:14:15,330 This total area under the probability density function must be equal to the total probability 116 00:14:15,330 --> 00:14:16,460 that is unity. 117 00:14:16,460 --> 00:14:21,260 So the total area under the probability density function, what is this quantity? 118 00:14:21,260 --> 00:14:33,850 This quantity is the total area 119 00:14:33,850 --> 00:14:48,490 under probability density function. 120 00:14:48,490 --> 00:14:54,540 The total area under the probability density function must be equal to 1. 121 00:14:54,540 --> 00:15:00,170 We are saying that the total area under the probability density function must be equal 122 00:15:00,170 --> 00:15:01,980 to 1. 123 00:15:01,980 --> 00:15:05,860 The probability density function is greater than or equal to 0, the total area under the 124 00:15:05,860 --> 00:15:08,950 probability density function is equal to 1. 125 00:15:08,950 --> 00:15:17,580 And, F of X times dX represents the probability that the random page table variable X takes 126 00:15:17,580 --> 00:15:21,170 a value in the infinitesimal interval of width dX around X. 127 00:15:21,170 --> 00:15:24,940 That is it lies in the interval between X to X plus dX. 128 00:15:24,940 --> 00:15:31,620 And integral A to B of F of X dX is basically the probability that the random variable takes 129 00:15:31,620 --> 00:15:38,140 values in the interval A to B. All right? 130 00:15:38,140 --> 00:15:45,520 So for example, let us take a look at this simple example now. 131 00:15:45,520 --> 00:16:11,870 For example, let us take a look at, consider the PDF, F of X of X equals K e to 132 00:16:11,870 --> 00:16:20,860 the power of minus AX for X greater than or equal to 0, that is we are given a PDF which 133 00:16:20,860 --> 00:16:27,800 is some constant, K times e raised to minus X for X greater than or equal to 0 where A 134 00:16:27,800 --> 00:16:29,290 is a known constant. 135 00:16:29,290 --> 00:16:32,920 A is also termed as the parameter. 136 00:16:32,920 --> 00:16:39,860 This quantity A is also termed as the parameter. 137 00:16:39,860 --> 00:16:53,050 A is the parameter of this PDF that is the probability density function. 138 00:16:53,050 --> 00:17:07,789 And what we are asked to do is we are asked to find the value of the unknown constant 139 00:17:07,789 --> 00:17:08,789 K. 140 00:17:08,789 --> 00:17:15,020 So we are given a PDF which is K e to the power of minus AX where A is a known parameter. 141 00:17:15,020 --> 00:17:21,549 This PDF is valid for X greater than or equal to 0 and what we are asked as we are asked 142 00:17:21,549 --> 00:17:24,639 what is the value of this constant, okay? 143 00:17:24,639 --> 00:17:26,769 And naturally, we can find it as follows. 144 00:17:26,769 --> 00:17:29,289 We know that the total probability is 1. 145 00:17:29,289 --> 00:17:34,749 Therefore, it must be the case that if I look at the total probability, this is defined 146 00:17:34,749 --> 00:17:36,820 from 0 to infinity. 147 00:17:36,820 --> 00:17:51,360 So therefore, if I look at the integral zero to infinity of F of X of X, for a general 148 00:17:51,360 --> 00:17:54,149 PDF it is minus infinity to infinity. 149 00:17:54,149 --> 00:17:56,429 Therefore, this must be equal to 1. 150 00:17:56,429 --> 00:18:01,759 However, this is defined only from 0 to infinity. 151 00:18:01,759 --> 00:18:05,750 It means that from minus infinity to infinity, the PDF is 0. 152 00:18:05,750 --> 00:18:18,080 So, 0 to infinity, K times e power minus AX dX is equal to 1. 153 00:18:18,080 --> 00:18:26,909 Now this integral, K times e power minus AX, you can evaluate this integral in a straightforward 154 00:18:26,909 --> 00:18:27,909 fashion. 155 00:18:27,909 --> 00:18:40,340 This is minus K over A e to the power of minus AX evaluated between the limits, 0 to infinity 156 00:18:40,340 --> 00:18:44,130 which must be equal to 1. 157 00:18:44,130 --> 00:18:49,940 It means, now if I substitute these limits at X equal to infinity, this is e power minus 158 00:18:49,940 --> 00:18:53,039 AX which is e power minus infinity. 159 00:18:53,039 --> 00:18:54,039 This is 0. 160 00:18:54,039 --> 00:19:01,230 Therefore, this is 0 minus of at X equal to 0, e power minus AX is e power minus0 which 161 00:19:01,230 --> 00:19:02,230 is one. 162 00:19:02,230 --> 00:19:12,860 So this is 0 minus of minus K by A which is equal to K by A and this is equal to 1 implies 163 00:19:12,860 --> 00:19:22,320 our parameter K is equal to A. So we derived that K by A is equal to 1 which means our 164 00:19:22,320 --> 00:19:26,559 parameter, remember over PDF is K e power minus AX. 165 00:19:26,559 --> 00:19:27,559 Right? 166 00:19:27,559 --> 00:19:31,529 Using the probability that the PDF must be integral to 1 that is the area under the PDF 167 00:19:31,529 --> 00:19:38,269 over the entire real line is 1, we have derived the condition that K is equal to A. 168 00:19:38,269 --> 00:19:55,820 Therefore the PDF is given as F of X of X which is equal to K is equal to 169 00:19:55,820 --> 00:20:04,590 A e to the power of minus AX for X greater than or equal to 0. 170 00:20:04,590 --> 00:20:17,499 So let me write this again clearly over here, F of X of X equals A e to the power of minus 171 00:20:17,499 --> 00:20:26,049 AX for X greater than or equal to 0 and this PDF which is decreasing exponentially, this 172 00:20:26,049 --> 00:20:28,570 PDF looks like this. 173 00:20:28,570 --> 00:20:34,720 This PDF for A is constant. 174 00:20:34,720 --> 00:20:38,820 It is decreasing exponentially. 175 00:20:38,820 --> 00:20:46,529 This is A e to the power of minus X, this is X. 176 00:20:46,529 --> 00:20:47,529 This is. 177 00:20:47,529 --> 00:20:49,809 F of X of X. 178 00:20:49,809 --> 00:20:51,090 It is decreasing. 179 00:20:51,090 --> 00:21:00,249 A e to the power of minus AX is basically decreasing exponentially. 180 00:21:00,249 --> 00:21:07,269 It starts with A at X is equal to 0 and it is decreasing exponentially. 181 00:21:07,269 --> 00:21:12,429 At X equal to 0, this is basically A e to the power of minus 0 which is A and it is 182 00:21:12,429 --> 00:21:20,419 decreasing exponentially, this PDF is also known as the exponential PDF. 183 00:21:20,419 --> 00:21:25,730 This is known as an exponential PDF. 184 00:21:25,730 --> 00:21:29,220 This is one of the standard probability density functions. 185 00:21:29,220 --> 00:21:32,960 This is known as an exponential probability density function. 186 00:21:32,960 --> 00:21:44,549 So F of X of X equals A e to the power of minus AX for X greater than or equal to 0. 187 00:21:44,549 --> 00:22:05,820 This is the exponential probability density function. 188 00:22:05,820 --> 00:22:12,929 This is the exponential probability density function and we have also seen that this A 189 00:22:12,929 --> 00:22:19,820 is a parameter which characterizes this exponential probability density function. 190 00:22:19,820 --> 00:22:25,509 So we are saying that this probability density function F of X of X equals A e to the power 191 00:22:25,509 --> 00:22:28,779 of minus AX is the exponential probability density function. 192 00:22:28,779 --> 00:22:32,950 It is defined for X greater than or equal to 0. 193 00:22:32,950 --> 00:22:37,999 And A is the parameter which characterizes this exponential probability density function. 194 00:22:37,999 --> 00:22:43,019 So in this module, we have seen a definition of the probability density function of a random 195 00:22:43,019 --> 00:22:46,869 variable, the properties of the probability density what it represents, the probability 196 00:22:46,869 --> 00:22:51,770 density function, the properties of this probability density function and also we have done a simple 197 00:22:51,770 --> 00:22:56,059 problem involving this exponential probability density function. 198 00:22:56,059 --> 00:22:57,419 So let us stop this module here. 199 00:22:57,419 --> 00:23:10,740 We will look at other aspects in the subsequent modules. 200 00:23:10,740 --> 00:23:26,340 Thank you very much