1 00:00:00,000 --> 00:00:00,190 2 00:00:00,190 --> 00:00:03,380 In this problem, we're given a random variable X which has a 3 00:00:03,380 --> 00:00:09,980 uniform distribution in the interval negative 1 to 1. 4 00:00:09,980 --> 00:00:15,030 In other words, if we were to draw out the PDF of X, we see 5 00:00:15,030 --> 00:00:19,290 that in the interval negative 1 to 1, it has value 1/2. 6 00:00:19,290 --> 00:00:22,460 7 00:00:22,460 --> 00:00:26,750 Now we're given a sequence random variables X1, X2, and 8 00:00:26,750 --> 00:00:32,380 so on, where each Xi has the same distribution as X and 9 00:00:32,380 --> 00:00:35,480 different Xi's are independent. 10 00:00:35,480 --> 00:00:38,820 For part a, we would like to know if the sequence Xi 11 00:00:38,820 --> 00:00:41,070 converges to some number-- 12 00:00:41,070 --> 00:00:42,390 let's call it c-- 13 00:00:42,390 --> 00:00:48,100 in probability as i goes to infinity-- 14 00:00:48,100 --> 00:00:49,710 whether this is true. 15 00:00:49,710 --> 00:00:52,030 Let's first recall the definition of convergence in 16 00:00:52,030 --> 00:00:53,410 probability. 17 00:00:53,410 --> 00:00:57,900 If this does happen, then by definition, we'll have that 18 00:00:57,900 --> 00:01:05,660 for every epsilon greater than 0, the probability Xi minus c 19 00:01:05,660 --> 00:01:11,510 greater equal to epsilon, this quantity will go to 0 in the 20 00:01:11,510 --> 00:01:15,490 limit of i going to infinity. 21 00:01:15,490 --> 00:01:20,450 In other words, with very high probability, we will find Xi 22 00:01:20,450 --> 00:01:24,220 to be very concentrated around the number c if this were to 23 00:01:24,220 --> 00:01:28,580 be the PDF of Xi. 24 00:01:28,580 --> 00:01:30,430 Now, can this be true? 25 00:01:30,430 --> 00:01:34,740 Well, we know that each Xi is simply a uniform distribution 26 00:01:34,740 --> 00:01:36,380 over negative 1 to 1. 27 00:01:36,380 --> 00:01:38,790 It doesn't really change as we increase i. 28 00:01:38,790 --> 00:01:41,420 So intuitively, the concentration around any 29 00:01:41,420 --> 00:01:43,840 number c is not going to happen. 30 00:01:43,840 --> 00:01:47,980 So we should not expect a convergence in probability in 31 00:01:47,980 --> 00:01:49,050 this sense. 32 00:01:49,050 --> 00:01:53,250 For part b, we would like to know whether the sequence Yi, 33 00:01:53,250 --> 00:01:58,370 defined as Xi divided by i, converges to anything in 34 00:01:58,370 --> 00:02:01,050 probability. 35 00:02:01,050 --> 00:02:04,250 Well, by just looking at the shape of Yi, we know that 36 00:02:04,250 --> 00:02:09,340 since the absolute value of Xi is less than 1, then we expect 37 00:02:09,340 --> 00:02:14,380 the absolute value of Yi is less than 1/i. 38 00:02:14,380 --> 00:02:17,400 So eventually, Yi gets very close to 0 39 00:02:17,400 --> 00:02:19,450 as i goes to infinity. 40 00:02:19,450 --> 00:02:23,650 So it's safe to bet that maybe Yi will converge to 0 in 41 00:02:23,650 --> 00:02:26,400 probability. 42 00:02:26,400 --> 00:02:28,870 Let's see if this is indeed the case. 43 00:02:28,870 --> 00:02:35,460 The probability of Yi minus 0 greater equal to epsilon is 44 00:02:35,460 --> 00:02:39,940 equal to the probability of Yi absolute value greater equal 45 00:02:39,940 --> 00:02:41,440 to epsilon. 46 00:02:41,440 --> 00:02:46,020 Now, previously we know that the absolute value of Yi is at 47 00:02:46,020 --> 00:02:49,270 most 1/i by the definition of Yi. 48 00:02:49,270 --> 00:02:53,000 And hence the probability right here is upper bounded by 49 00:02:53,000 --> 00:02:57,370 the probability of 1i greater equal to epsilon. 50 00:02:57,370 --> 00:02:59,740 Notice in this expression, there is nothing random. 51 00:02:59,740 --> 00:03:01,410 i is simply a number. 52 00:03:01,410 --> 00:03:11,245 Hence this is either 1 if i is less equal to 1/epsilon, or 0 53 00:03:11,245 --> 00:03:15,700 if i is greater than 1/epsilon. 54 00:03:15,700 --> 00:03:20,520 Now, this tells us, as long as i is great enough-- it's big 55 00:03:20,520 --> 00:03:22,720 enough compared to epsilon-- 56 00:03:22,720 --> 00:03:24,720 we know that this quantity here is [INAUDIBLE] 57 00:03:24,720 --> 00:03:25,500 0. 58 00:03:25,500 --> 00:03:30,570 And that tells us in the limit of i goes to infinity 59 00:03:30,570 --> 00:03:35,910 probability of Yi deviating from 0 by more than 60 00:03:35,910 --> 00:03:40,070 epsilon goes to 0. 61 00:03:40,070 --> 00:03:47,080 And that shows that indeed, Yi converges to 0 in probability 62 00:03:47,080 --> 00:03:51,100 because the expression right here, this limit, holds for 63 00:03:51,100 --> 00:03:53,180 all epsilon. 64 00:03:53,180 --> 00:03:55,400 Now, in the last part of the problem, we are looking at a 65 00:03:55,400 --> 00:04:01,100 sequence Zi defined by Xi raised to the i-th power. 66 00:04:01,100 --> 00:04:03,940 Again, since we know Xi is some number between negative 1 67 00:04:03,940 --> 00:04:07,380 and 1, this number raised to the i-th power is likely to be 68 00:04:07,380 --> 00:04:08,630 very small. 69 00:04:08,630 --> 00:04:10,760 And likely to be small in the sense that it will have 70 00:04:10,760 --> 00:04:12,830 absolute value close to 0. 71 00:04:12,830 --> 00:04:16,660 So a safe guess will be the sequence Zi converges to 0 as 72 00:04:16,660 --> 00:04:20,500 well as i goes to infinity. 73 00:04:20,500 --> 00:04:22,490 How do we prove this formally? 74 00:04:22,490 --> 00:04:26,810 We'll start again with a probability that Zi stays away 75 00:04:26,810 --> 00:04:31,970 from 0 by more than epsilon and see how that evolves. 76 00:04:31,970 --> 00:04:36,770 And this is equal to the probability that Xi raised to 77 00:04:36,770 --> 00:04:40,920 the i-th power greater equal to epsilon. 78 00:04:40,920 --> 00:04:46,330 Or again, we can write this by taking out the absolute value 79 00:04:46,330 --> 00:04:51,410 that Xi is less equal to negative epsilon raised to the 80 00:04:51,410 --> 00:04:57,970 1 over i-th power or Xi greater equal to epsilon 1 81 00:04:57,970 --> 00:05:01,470 over i-th power. 82 00:05:01,470 --> 00:05:05,610 So here, we'll divide into two cases, depending on the value 83 00:05:05,610 --> 00:05:07,000 of epsilon. 84 00:05:07,000 --> 00:05:10,670 In the first case, epsilon is greater than 1. 85 00:05:10,670 --> 00:05:15,130 Well, if that's the case, then we know epsilon raised to some 86 00:05:15,130 --> 00:05:18,590 positive power is still greater than 1. 87 00:05:18,590 --> 00:05:22,760 But again, Xi cannot have any positive density be on the 88 00:05:22,760 --> 00:05:25,470 interval negative 1 or 1. 89 00:05:25,470 --> 00:05:28,650 And hence we know the probability above, which is Xi 90 00:05:28,650 --> 00:05:31,780 less than some number smaller than negative 1 or greater 91 00:05:31,780 --> 00:05:34,720 than some number bigger than 1 is 0. 92 00:05:34,720 --> 00:05:36,910 So that case is handled. 93 00:05:36,910 --> 00:05:40,210 Now let's look at a case where epsilon is less than 1, 94 00:05:40,210 --> 00:05:41,870 greater than 0. 95 00:05:41,870 --> 00:05:47,010 So in this case, epsilon to the 1/i will be less than 1. 96 00:05:47,010 --> 00:05:50,380 And it's not that difficult to check that since Xi has 97 00:05:50,380 --> 00:05:55,280 uniform density between negative 1 and 1 of magnitude 98 00:05:55,280 --> 00:06:02,290 1/2, then the probability here was simply 2 times 1/2 times 99 00:06:02,290 --> 00:06:07,590 the distance between epsilon to the 1 over 100 00:06:07,590 --> 00:06:11,390 i-th power and 1. 101 00:06:11,390 --> 00:06:15,080 So in order to prove this quantity converge to 0, we 102 00:06:15,080 --> 00:06:19,200 simply have to justify why does epsilon to the 1/i 103 00:06:19,200 --> 00:06:23,570 converge to 1 as i goes to infinity. 104 00:06:23,570 --> 00:06:25,120 For that, we'll recall the properties 105 00:06:25,120 --> 00:06:26,560 of exponential functions. 106 00:06:26,560 --> 00:06:30,910 In particular, if a is a positive number and x is its 107 00:06:30,910 --> 00:06:35,920 exponent, if we were to take the limit as x goes to 0 and 108 00:06:35,920 --> 00:06:39,370 look at the value of a to the power of x, we see that 109 00:06:39,370 --> 00:06:41,190 this goes to 1. 110 00:06:41,190 --> 00:06:46,920 So in this case, we'll let a be equal to epsilon and x be 111 00:06:46,920 --> 00:06:48,640 equal to 1/i. 112 00:06:48,640 --> 00:06:52,450 As we can see that as i goes to infinity, the value of x, 113 00:06:52,450 --> 00:06:55,070 which is 1/i, does go to 0. 114 00:06:55,070 --> 00:06:59,540 And therefore, in the limit i going to infinity, the value 115 00:06:59,540 --> 00:07:04,500 of epsilon to the 1 over i-th power goes to 1. 116 00:07:04,500 --> 00:07:09,380 And that shows if we plug this limit into the expression 117 00:07:09,380 --> 00:07:13,860 right here that indeed, the term right here goes to 0 as i 118 00:07:13,860 --> 00:07:15,150 goes to infinity. 119 00:07:15,150 --> 00:07:20,740 And all in all, this implies the probability of Zi minus 0 120 00:07:20,740 --> 00:07:25,040 absolute value greater equal to epsilon in the limit of i 121 00:07:25,040 --> 00:07:30,370 going to infinity converges to 0 for all positive epsilon. 122 00:07:30,370 --> 00:07:34,230 And that completes our proof that indeed, Zi converges to 0 123 00:07:34,230 --> 00:07:35,480 in probability. 124 00:07:35,480 --> 00:07:36,080