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Title: Properties of Variance and Standard Deviation
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Series: Probability Theory
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YouTube-Title: Probability Theory 18 | Properties of Variance and Standard Deviation
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Bright video: https://youtu.be/f0G2nZQxe08
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Dark video: https://youtu.be/v8LY_xwsVnk
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: pt18_sub_eng.srt
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Timestamps
00:00 Intro
00:35 Properties
01:30 Variance is additive
01:50 Scaling variance
02:20 Scaling standard deviation
03:04 Proof of properties
07:48 Credits
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Subtitle in English
1 00:00:00,371 –> 00:00:04,039 Hello and welcome back to probability theory.
2 00:00:04,239 –> 00:00:11,769 We’ve already reached part 18 in the series and we are still talking about the variance and the standard deviation.
3 00:00:11,969 –> 00:00:17,678 More precisely today we will prove some important formulas that hold for independent random variables.
4 00:00:18,814 –> 00:00:23,604 However before we start with this, a big thank you to all my supporters on Steady.
5 00:00:24,029 –> 00:00:28,927 You can also support this channel via PayPal or by other means, mentioned in the description
6 00:00:29,300 –> 00:00:34,653 and as a supporter, you have access to PDF versions and quizzes for all the videos.
7 00:00:34,853 –> 00:00:37,556 Ok, then let’s start with the topic of today.
8 00:00:37,756 –> 00:00:41,799 Let’s talk about some properties of the variance and the standard deviation
9 00:00:41,999 –> 00:00:47,943 and as I said before, we will formulate this for 2 independent random variables X and Y.
10 00:00:48,514 –> 00:00:53,440 Moreover we already know from the last videos, that the variance is only well defined,
11 00:00:53,514 –> 00:00:57,183 if the integral of the random variables squared exists
12 00:00:57,886 –> 00:01:03,391 or in other words the expectation of X squared has to be a finite number
13 00:01:03,591 –> 00:01:07,014 and of course the same we also need for the random variable Y.
14 00:01:07,071 –> 00:01:11,533 Which means the expectation of Y squared has to be a finite number as well.
15 00:01:11,733 –> 00:01:17,870 OK, with this we now know the variance and the standard deviation of X and Y exist.
16 00:01:18,471 –> 00:01:21,786 So all of these are well defined real numbers.
17 00:01:22,286 –> 00:01:27,933 However, now we are also able to say something about the sum of the 2 random variables.
18 00:01:28,133 –> 00:01:32,973 More concretely we can now calculate the variance of X + Y
19 00:01:33,173 –> 00:01:38,431 and maybe it’s not a big surprise for you. In this case the variance is indeed additive.
20 00:01:39,000 –> 00:01:42,653 This means we can simply pull out the plus sign here.
21 00:01:43,186 –> 00:01:48,544 However here please don’t forget, it’s important that the two random variables here are independent.
22 00:01:49,157 –> 00:01:55,209 OK, so this is the addition and now you might ask: what happens when we scale a random variable?
23 00:01:55,409 –> 00:02:01,230 So the question is what is the variance of Lambda times X, where Lambda is a real number
24 00:02:01,743 –> 00:02:05,799 and in fact, we can pull out this Lambda factor, but with a square.
25 00:02:06,214 –> 00:02:11,538 So we can simplify our variance here, but we shouldn’t forget this square outside.
26 00:02:11,738 –> 00:02:18,169 OK, now you might have the idea, that we should formulate the second rule here for the standard deviation as well,
27 00:02:18,369 –> 00:02:21,794 because there the square should vanish in the definition.
28 00:02:21,994 –> 00:02:29,627 There please recall, the standard deviation Sigma of Lambda times X, is simply the square root of the variance.
29 00:02:29,827 –> 00:02:34,357 This means indeed here on the right-hand side the square of Lambda vanishes,
30 00:02:34,358 –> 00:02:36,568 but now we have the absolute value there.
31 00:02:36,768 –> 00:02:41,708 This makes sense, because the standard deviation has to be always positive.
32 00:02:42,286 –> 00:02:47,582 Hence also here you should remember, you can pull out a scalar from the standard deviation,
33 00:02:47,614 –> 00:02:50,390 but then you should forget about the minus signs.
34 00:02:51,029 –> 00:02:56,519 In other words if you just look at positive number, this is a homogeneous property.
35 00:02:57,171 –> 00:03:03,146 Now, I would say all the 3 rules here, you should remember, because you can use them in calculations.
36 00:03:03,346 –> 00:03:06,819 However, of course first we should prove them.
37 00:03:07,300 –> 00:03:09,492 Indeed this is not a hard proof,
38 00:03:09,500 –> 00:03:14,786 because we can immediately use the formula we have for the variance given by the expectations.
39 00:03:15,400 –> 00:03:20,935 This means the variance of X + Y is given by the difference of two expectations.
40 00:03:21,343 –> 00:03:27,981 Please recall in the first one we have the square inside the expectation and in the second one outside
41 00:03:28,271 –> 00:03:35,440 and now the good thing we can use here is, that the properties of the expectation, we have already discussed in part 15.
42 00:03:35,640 –> 00:03:43,534 In particular we know the expectation is linear and of course this should be something that can help us here.
43 00:03:44,329 –> 00:03:49,402 However first here, let’s expand the square inside the expectation.
44 00:03:49,771 –> 00:03:54,725 So this is X squared + 2 times X times Y + Y squared.
45 00:03:54,925 –> 00:03:59,547 So not so complicated and in the next step we can use the linearity here.
46 00:03:59,747 –> 00:04:04,286 Moreover here on the right-hand side, we can use the linearity immediately.
47 00:04:04,486 –> 00:04:12,169 So we have the expectation of X + the expectation of Y and still we have to square the whole thing.
48 00:04:12,700 –> 00:04:18,424 Ok, then in the next step let’s use the linearity of the expectation here in the first part.
49 00:04:18,814 –> 00:04:22,190 So we can pull out the additions and the scalars.
50 00:04:22,390 –> 00:04:33,654 Hence first we have expectation of X squared + 2 times the expectation of X times Y + the expectation of Y squared.
51 00:04:33,854 –> 00:04:39,891 Ok, then on the other hand here the second part we can expand the square as well.
52 00:04:40,091 –> 00:04:43,941 So this means first we have the expectation of X squared,
53 00:04:44,600 –> 00:04:53,896 then minus 2 times the expectation of X times the expectation of Y and finally we have this squared expectation of Y.
54 00:04:54,529 –> 00:05:01,850 Ok at this point you should see, we can already put this term together with this term to get the variance of X
55 00:05:02,171 –> 00:05:09,727 and in the same sense this expectation here and that expectation there gives us the variance of Y.
56 00:05:09,927 –> 00:05:14,536 So you see we have already reached what we want here on the right-hand side.
57 00:05:15,014 –> 00:05:19,950 Therefore the only question is what is with the 2 remaining terms here?
58 00:05:20,443 –> 00:05:23,966 So maybe first to answer this let’s put them together.
59 00:05:24,629 –> 00:05:31,643 So it’s two times then the expectation of X times Y minus the expectation of X times the expectation of Y.
60 00:05:31,843 –> 00:05:35,725 So at this point we know this whole term should be 0
61 00:05:36,157 –> 00:05:41,690 and indeed there the independence of the 2 random variables X and Y comes in,
62 00:05:41,890 –> 00:05:49,490 because if they are independent, then this expectation here is just the product of 2 expectations.
63 00:05:49,986 –> 00:05:53,704 That’s also something we have stated in part 15.
64 00:05:54,214 –> 00:06:03,860 So the conclusion here is, the whole term inside the parentheses here is 0 and this finishes the proof of part (a).
65 00:06:04,286 –> 00:06:12,737 So now let’s go to part (b). In fact, part (b) is much easier, because only one random variable is involved here,
66 00:06:13,229 –> 00:06:17,517 but again we can use the formula with the expectations for the variance.
67 00:06:17,971 –> 00:06:23,901 So first we have the expectation, where the random variables is squared and then we square the expectation
68 00:06:24,101 –> 00:06:28,838 and now you should see in both terms we can pull out a Lambda squared.
69 00:06:29,271 –> 00:06:33,442 Again simply by using the linearity of the expectation.
70 00:06:33,642 –> 00:06:38,746 Ok and then the only thing we have to do now is to factor out the Lambda squared,
71 00:06:39,057 –> 00:06:43,490 because then only the difference of the two expectations remains
72 00:06:43,690 –> 00:06:49,791 and we have used this formula so often now, that you immediately see this is the variance of X.
73 00:06:49,991 –> 00:06:59,269 In other words we have proven part (b) and also not a surprise at all from part (b), we now can derive part (c).
74 00:06:59,469 –> 00:07:05,676 Essentially part (c) was the same thing, but now for the standard deviation instead of the variance.
75 00:07:06,200 –> 00:07:10,850 Hence the only thing we have to do here, is to take the square root of the variance
76 00:07:11,271 –> 00:07:16,349 and then by using part (b), we see we have the square root of Lambda squared
77 00:07:16,843 –> 00:07:21,329 and indeed this is simply now the absolute value of Lambda
78 00:07:21,471 –> 00:07:27,716 and the other factor is just this square root of the variance of X, which is the standard deviation of X.
79 00:07:28,129 –> 00:07:32,241 Ok and with this you see the whole proof is finished.
80 00:07:32,441 –> 00:07:39,494 So as I said please remember all these properties, all the formulas here, because you can use them in calculations
81 00:07:40,114 –> 00:07:43,786 and indeed how to do that, we will see in the next videos.
82 00:07:43,871 –> 00:07:48,143 Therefore I really hope that I see you there and have a nice day. Bye!
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Quiz Content
Q1: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. What is correct for the variance and $\lambda \in \mathbb{R}$?
A1: $\mathrm{Var}(\lambda X) = \lambda^2 \mathrm{Var}(X)$
A2: $\mathrm{Var}(\lambda X) = \lambda \mathrm{Var}(X)$
A3: $\mathrm{Var}(\lambda X) = \lambda^3 \mathrm{Var}(X)$
A4: $\mathrm{Var}(\lambda^2 X) = \lambda \mathrm{Var}(X)$
Q2: Let $X, Y$ be two independent random variables. What is correct for $X+Y$ in general?
A1: $\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) $
A2: $\mathrm{Var}(X + Y) = \mathrm{Var}(X) - \mathrm{Var}(Y) $
A3: $\sigma(X + Y) = \sigma(X) + \sigma(Y) $
A4: $\sigma(X + Y) = \sigma(X) \cdot \sigma(Y) $
Q3: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. What is correct for the standard deviation and for every $\lambda \in \mathbb{R}$?
A1: $\sigma(-X) = \sigma(X)$
A2: $\sigma(2 X) = 4 \sigma(X)$
A3: $\sigma(\lambda X) =\lambda^2 \sigma(X) $
A4: $\sigma(\lambda X) =\lambda \sigma(X) $
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Last update: 2024-10
