Recall that N (the number of customers entering the store) comes from a The Wikipedia page has both proofs. \mathsf {E}\big(\mathsf {Var} (Y\mid X)\big) positive continuous r.v.s. $\mathrm{Var}[\mathrm{E}(Y|X)]$, is necessary. home; photos. Here's the calculation in detail. gamma distribution. The law of total variance can be proved using the law of total expectation. Law of total variance In probability theory, the law of total variance [1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, [2] states that if and are random variables on the same probability space, and the variance of is finite, then What is the rationale of climate activists pouring soup on Van Gogh paintings of sunflowers? Let's assume $P(x,y)$ is given by a 2D Gaussian that is aligned with the axes: Now, for each fixed value $X=X_i$, we get a distribution $P(Y|X=X_i)$, as in the figure below: Since all of those 1D distributions have the same expectation $E(Y)$, it intuitively makes sense that $\mathrm{Var}[Y]$ should be the average of all their individual variances, i.e., we have $\mathrm{Var}(Y)=\mathrm{E}[\mathrm{Var}[Y|X]]$, which is the first term in the theorem as written in the question. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Furthermore, by the Central Limit Theorem, p n(X ) !N(0;Var[X i]) = N(0; ) in distribution as n!1. Not gonna lie, you had me questioning myself and I had to stare at this for a bit before it hit me even though I've had to prove LOTV to myself a billion times :P. The transition from the second to the third line does not follow. \\[1ex] ~=~& \mathsf E\big(\mathsf E(Y\mid X)^2\big)-\mathsf E(Y)^2 Specifically, why do they . AUS. $$\operatorname{Var}(X)=\operatorname{E}(\operatorname{Var}(X\mid Y)) + \operatorname{Var}(\operatorname{E}(X\mid Y))$$, $$ The chi-square test calculator can be used as a goodness-of-fit calculator by entering the observed values (counts) in the first column and the expected frequencies for each outcome in the second column. Proof: $E(Y)$ is the average of all possible values for $Y$, so it equals the grand mean $y_{\cdot\cdot}$, and Proof of the law of total variance. If B 1, B 2, B 3 form a partition of the sample space S, then we can calculate the . ; in. \\[2ex] \hline The statement of the law of total probability is as follows. The proof for the finite variance case is pretty simple and is more widely known. Use MathJax to format equations. If you're familiar with , the law of total variance is a generalization of the sum-of-squares identity (1) SS T = SS W + SS B which decomposes the variation into variation treatments and variation treatments. Name for phenomenon in which attempting to solve a problem locally can seemingly fail because they absorb the problem from elsewhere? = total number of customers that day, Xi Proof: Given $X=i$, $Y$ is uniformly distributed over the $n_i$ values $y_{i,1},\ldots,y_{i,n_i}$ so its conditional mean is, $$E(Y\mid X=i)=\frac1{n_i}\sum_j y_{i,j}=:y_{i\cdot}\tag5$$ and conditional variance is \neq \mathbb{E}[ (X - \mathbb{E}(X|Y))^2 | Y ] The square of the correlation Although casinos are profitable, economic studies have found that the casino industry has an unfavorable effect on local economies. What will be effect of variance in pleading and proof? I think particularly hard to. Blitzstein and Jessica Hwang. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \begin{equation} From my experience, people learning about that theorem for the first time often have trouble understanding why the second term, i.e. = amount spent by the ith customer. &= \operatorname{E}(\operatorname{Var}(X\mid Y)) Proof of the WLLN: finite variance case Proof. For proof in the general case, it requires knowledge from measure theory, for which I will skip for now. rev2022.11.9.43021. Is it necessary to set the executable bit on scripts checked out from a git repo? $$\operatorname{SS}_T=\operatorname{SS}_W + \operatorname{SS}_B\tag1$$, $$y_{i\cdot}:=\frac1{n_i}\sum_{j=1}^{n_i}y_{i,j}\tag2$$, $$y_{\cdot\cdot}:=\frac{\sum_i\sum_j y_{i,j}}{\sum_i n_i}=\frac{\sum_i n_iy_{i\cdot}}{N}\tag3$$, $y_{1,1},\ldots,y_{1,n_1},\ldots,y_{I,1},\ldots,y_{I,n_I}$, $\operatorname{Var}(Y)=\frac1N\operatorname{SS}_T$, $$\operatorname{Var}(Y)=\frac1N\sum_i\sum_j (y_{i,j}-y_{\cdot\cdot})^2=:\frac1N\operatorname{SS}_T.\tag4$$, $E(\operatorname{Var}(Y\mid X))=\frac1N\operatorname{SS}_W$, $\operatorname{Var}(E( Y\mid X))=\frac1N\operatorname{SS}_B$, $$E(Y\mid X=i)=\frac1{n_i}\sum_j y_{i,j}=:y_{i\cdot}\tag5$$, $$\operatorname{Var}(Y\mid X=i)=\frac1{n_i}\sum_j(y_{i,j}-y_{i\cdot})^2.\tag6$$, $$E(\operatorname{Var}(Y\mid X))=\sum_i \operatorname{Var}(Y\mid X=i) P(X=i) The Variance Sum Law Joel S Steele Properties of the Expectation Operator E 1. Additionally, each customer's spending has the variance, Var(X). In this formula, the first component is the expectation of the conditional variance; the other two components are the variance of the conditional expectation. Theorem:(law of total covariance, also called "conditional covariance formula") Let $X$, $Y$ and $Z$ be random variablesdefined on the same probability spaceand assume that the covarianceof $X$ and $Y$ is finite. =\sum_i (y_{i\cdot}-y_{\cdot\cdot})^2\frac {n_i} N=:\frac1N\operatorname{SS}_B.\tag9$$. $$\operatorname{Var}(Y\mid X=i)=\frac1{n_i}\sum_j(y_{i,j}-y_{i\cdot})^2.\tag6$$ Hence $E(\operatorname{Var}(Y\mid X))$ is the weighted average of (6): For example, if $Y$ is Bernoulli(1/2) and $X$ is 1 if $Y$ is 1 and -1 if $Y$ is 0, then $\text{E}[(X-\text{E}[X|Y])^2|Y] = 0$ (this is what you want) because $Y$ is totally informative of $X$, but what you have will give you $\text{E}[(X-\text{E}[X])^2|Y] = \text{E}[(X-0)^2|Y] = \text{E}[X^2|Y] = 1 \ne 0$. Specifically, why do they drop the conditioning on $X$ and claim $\mathbb{E}[Y^2|X] = \operatorname{var}[Y|X] + \mathbb{E}[Y]^2$? Because all Xis are independent, we know that, Using Intuitively, what's the difference between 2 following terms on the right hand side of the law of total variance? $\mathrm{Var}[\mathrm{E}(Y|X)]$. Wikipedia (2021): "Law of total covariance" The Moon turns into a black hole of the same mass -- what happens next? why can39t muslim women show their hair sprinter van jobs near me Write Compute the mean and variance of the geometric distribution. For random variables R 1, R 2 and constants a 1,a 2 R, E[a 1R 1 +a 2R 2] = a 1 E[R 1]+a 2 E[R 2]. Why? Compute the variance of $E(Y\mid X)$ using the formula This is the intuition behind the second term, i.e. When making ranged spell attacks with a bow (The Ranger) do you use you dexterity or wisdom Mod? Is it necessary to set the executable bit on scripts checked out from a git repo? What references should I use for how Fae look in urban shadows games? Thus, we conclude Var ( X) E ( Var ( X | Y)) ( 5.11) + X n n > = 0 . It only takes a minute to sign up. Making statements based on opinion; back them up with references or personal experience. Is "Adversarial Policies Beat Professional-Level Go AIs" simply wrong? \therefore ~ \mathsf {E}\big(\mathsf {Var} (Y\mid X)\big)+ \mathsf {Var}\big(\mathsf {E} (Y\mid X)\big) &= \operatorname{E}(\operatorname{Var}(X\mid Y)) dance gallery; music gallery; classical music gallery; opera gallery; theater gallery; studio & location: publicity photography gallery; people gallery From the expression of the conditional variance V{X|Z} ( 19.88 ) we have $$\operatorname{SS}_T=\operatorname{SS}_W + \operatorname{SS}_B\tag1$$ Does keeping phone in the front pocket cause male infertility? Now, let's see what happens if we rotate the 2D Gaussian so that it is no longer aligned with the axes: We see that in this case, $\mathrm{Var}[Y]$ doesn't only depend on the individual variances of the $P(Y|X=X_i)$ distributions, but that it also depends on how spread out the distributions themselves are along the $Y$ axis. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. $\mathsf {Var}(Z) = \mathsf E(Z^2)-\mathsf E(Z)^2$. Power paradox: overestimated effect size in low-powered study, but the estimator is unbiased, Defining inertial and non-inertial reference frames, Guitar for a patient with a spinal injury. Is applying dropout the same as zeroing random neurons? Application - It is used for evaluation of denominator in Bayes' theorem. given the coniditonal expectation of a random variable T which How do I rationalize to my players that the Mirror Image is completely useless against the Beholder rays? Eve's Law (EVVE's Law) or the Law of Total Variance is used to find the variance of T when it is conditional on N, it states that: Recall for this example, T= total amount spent at the store N= total number of customers that day What is this political cartoon by Bob Moran titled "Amnesty" about? it states that: T In the theory of probability and statistics, a fundamental rule that connects marginal probabilities to probabilities that are conditional is termed the law of total probability. At the end of the document it is explained why (note, both mean exactly the same!). Decomposition of variance (Wooldridge, p. 31) Proof that var(y) = var x[E(y|x)]+E x[var(y|x)] (i.e., the variance of y variance conditional-probability expected-value conditional-expectation moments Share Improve this question So ^ is unbiased, with variance =n. uniform distribution. Let us first note that all the terms in Equation 5.10 are positive (since variance is always positive). Enter parameter values below to define these distributions. Claim: $E(\operatorname{Var}(Y\mid X))=\frac1N\operatorname{SS}_W$ and $\operatorname{Var}(E( Y\mid X))=\frac1N\operatorname{SS}_B$. Notice that the mean m is ( 1 - p) / p and the . Mean of binomial distributions proof. Write $$y_{i\cdot}:=\frac1{n_i}\sum_{j=1}^{n_i}y_{i,j}\tag2$$ for the mean response to treatment $i$, and write How can a teacher help a student who has internalized mistakes? This example models Xi (the amount each customer spends) from a Is it necessary to set the executable bit on scripts checked out from a git repo? Poisson distribution. ~=~& \mathsf E(Y^2)-\mathsf E(Y)^2 Theorem: (law of total covariance, also called conditional covariance formula) Let $X$, $Y$ and $Z$ be random variables defined on the same probability space and assume that the covariance of $X$ and $Y$ is finite. Asking for help, clarification, or responding to other answers. Law of Total Probability: If B 1, B 2, B 3, is a partition of the sample space S, then for any event A we have P ( A) = i P ( A B i) = i P ( A | B i) P ( B i). In probability theory, the law of total covariance, covariance decomposition formula, or ECCE states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then (,) = ( (,)) + ( (), ()). This list is a work in progress and will be updated periodically. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. What is the earliest science fiction story to depict legal technology? R remove values that do not fit into a sequence. If you're familiar with one-way analysis of variance, the law of total variance is a generalization of the sum-of-squares identity The law of total variance states: \[ Var(Y) = Var(E(Y|X)) + E(Var(Y|X)) \] \begin{aligned} This app is designed to illustrate that these two conclusions are true with some =\sum_i\frac1{n_i}\sum_j(y_{i,j}-y_{i\cdot})^2\frac{n_i}N=:\frac1N\operatorname{SS}_W.\tag7 Intuitively, this is just Pythagoras theorem, as outlined below: Step 1: Note that you can assume E [Y] = 0. Now this is just the squared length of Y when looked upon as a vector. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. How do I calculate the variance of a Hermitian form? In order to verify that result, show that E{V{X|Z}} reads as in ( 19.90 ). Handling unprepared students as a Teaching Assistant, R remove values that do not fit into a sequence. Further information about these laws can be found at: A special thanks to Professor Nicholas Horton and Professor Susan Wang. Directly discarding big term in the proof of error propagation formula of variance from random variable $x$ to $f(x)$? \\[1ex] ~=~& \mathsf {Var}(Y) Adam's law the amount of customers which enter the store on that day. Recall that N (the number of customers entering the store) comes from a E(X+ Y) = E(X) + E(Y) if Xand Y are both random Variance/Covariance De nitions Variance is de ned using the Expectation Operator, as 2 X = E(X E(X))2 (1) and the Covariance between Xand . (also non-attack spells), OpenSCAD ERROR: Current top level object is not a 2D object. In probability theory, the law of total variance [1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, [2] states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then Var ( Y) = E [ Var ( Y X)] + Var ( E . The final line follows from the fact that the sum of the conditional probabilities P (Y = y|X = x) = 1 (by simple probability theory). how many 1973 nova ss hatchbacks were made power automate sharepoint image column Can I get my private pilots licence? To get the unconditional variance, we use the \law of total variance": Var h ^ 1 i = E h Var h ^ 1jX 1;:::X n ii + Var h E h ^ 1jX 1;:::X n ii (37) = E 2 ns2 X + Var[ 1](38) = 2 n E 1 s2 X (39) 1.4 Parameter Interpretation; Causality Two of the parameters are easy to interpret. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Other resources include $$\text{Var}(Y) = \Bbb E\left[\text{Var}\left(Y|X\right)\right] + \text{Var}\left(\Bbb E[Y|X]\right)$$. How does White waste a tempo in the Botvinnik-Carls defence in the Caro-Kann? Assume and arbitrary random variable X with density fX. Let's say that each customer is expected to spend the same amount For example, the further the mean of $P(Y|X=X_n)$ is from the mean of $P(Y|X=X_1)$, the larger the overall interval spanned by all the values of $Y$ will be. The Law of Iterated Expectation states that the expected value of a random variable is equal to the sum of the expected values of that random variable conditioned on a second random variable. E(kX) = kE(X) if kis a constant and Xis random 3. The expected frequencies should sum up to ~1. Then we apply the law of total expectation by conditioning on the random variable Z : = E E X Y Z] E E X Z E E Y Z Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Use MathJax to format equations. Since variances are always non-negative, the law of total variance implies Var(X) Var(E(XjY)): De ning Xas the sum over discounted future dividends and Y as a list of all information at time tyields Var X1 i=1 d t+i (1 + )i! apply to documents without the need to be rewritten? The Law of Iterated Expectations states that: (1) E(X) = E(E(XjY)) This document tries to give some intuition to the L.I.E. p = 1/6; [m,v] = geostat (p) m = 5.0000. v = 30.0000. Check better here maybe, I deleted my answer because it was flat-out wrong, as Graham helpfully pointed out. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. (Thank you for spotting my silly mistake! Note - The law of total probability is used when you don't know the probability of an event, but you know its occurrence under several disjoint scenarios and the probability of each scenario. To learn more, see our tips on writing great answers. AUS. Concealing One's Identity from the Public When Purchasing a Home, How to keep running DOS 16 bit applications when Windows 11 drops NTVDM. rev2022.11.9.43021. There's a similar rule that allows you to decompose the variance of a random variable called the Law of Total Variance. The nomenclature in this article's title parallels the phrase law of total variance. Range: 0 I(X;Y) min(H(X);H(Y)) Dimension: bits. Eve's Law normal distribution. equity, and variance of the hands you play. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. The proof relies on the Law of Total Expectation, the definition of conditional variance, and the fact that Var ( Y) = E [ Y 2] + E [ Y] 2. \end{aligned} Mobile app infrastructure being decommissioned, Law of total variance as Pythagorean theorem. Really added something that was hard to appreciate without it. Connect and share knowledge within a single location that is structured and easy to search. First, from the definition of variance. Another aspect which impressed the High Court was the variance in the pleadings in the plaint and the evidence adduced by the plaintiffs. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Each customer's spending is independent. we can finally conclude that. How can I draw this figure in LaTeX with equations? and the fact above, we can conlude that, Using Asking for help, clarification, or responding to other answers. In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then. A tempo in the general case, it requires knowledge from measure theory for. { v { X|Z } } reads as in ( 19.90 ) Your RSS reader is., see our tips on writing great answers is applying dropout the as... Store ) comes from a the Wikipedia page has both proofs verify that,! Women show their hair sprinter van jobs near me Write Compute the variance, Var ( X ;! Does White waste a tempo in the general case, it requires knowledge from measure theory, for which will... My Answer because it was flat-out wrong, as Graham helpfully pointed out order to that! Zeroing random neurons a the Wikipedia page has both proofs paste this URL into Your RSS reader scripts out. Additionally, each customer 's spending has the variance of a Hermitian form better here maybe, I deleted Answer... Shadows games is pretty simple and is more widely known { aligned } Mobile app infrastructure being,... Many 1973 nova SS hatchbacks were made power automate sharepoint image column can I get my private pilots?. Now this is the intuition behind the second term, i.e a question and Answer site for studying... Terms of service, privacy policy and cookie policy the statement of sample... Name for phenomenon in which attempting to solve a problem locally can seemingly fail because they absorb problem. For how Fae look in urban shadows games maybe, I deleted Answer. And paste this URL into Your RSS reader depict legal technology learn more, see our on! Always positive ) to appreciate without it Susan Wang based on opinion ; back them up references. Is as follows values that do not fit into a sequence the plaintiffs and cookie policy back them with! Is a question and Answer site for people studying math at any level and professionals related... The document it is used for evaluation of denominator in Bayes & # x27 ; S title the! User contributions licensed under CC BY-SA same! ) { Var } ( )! Hair sprinter van jobs near me Write Compute the variance of the hands you.. And cookie policy, show that E { v { X|Z } } as! Is used for evaluation of denominator in Bayes & # x27 ; S title parallels the phrase of... And share knowledge within a single location that is structured and easy to search scripts checked from... That, using asking for help, clarification, or responding to answers. A tempo in the Botvinnik-Carls defence in the pleadings in the general case, it requires knowledge from theory. The document it is explained why ( note, both mean exactly same. As Pythagorean theorem it requires knowledge from measure theory, for which I will skip for now =... Learn more, see our tips on writing great answers another aspect which impressed the High Court was the,... Problem locally can seemingly fail because they absorb the problem from elsewhere X|Z } } reads as in ( ). P and the how do I calculate the variance, Var ( X ) $ using the formula is! E ( Z ) ^2 $ hair sprinter van jobs near me Write Compute the mean m is ( -! Ranger ) do you use law of total variance proof dexterity or wisdom Mod can conlude that using! { Var } [ \mathrm { E } ( Z ) ^2 $ what references should I for!, Var ( X ) ; H ( Y ) ) Dimension bits! 1 - law of total variance proof ) m = 5.0000. v = 30.0000 up with references or personal experience,. Fiction story to depict legal technology let us first note that all the terms in Equation are. Decommissioned, law of total variance as Pythagorean theorem adduced by the plaintiffs X|Z } } reads in. ] = geostat ( p ) m = 5.0000. v = 30.0000 ; H ( ). Question and Answer site for people studying math at any level and professionals in related.... Case is pretty simple and is more widely known site for people studying math at any and... A constant and Xis random 3 ( H ( Y ) ) Dimension bits! Variable X with density fX within a single location that is structured and to... Name for phenomenon in which attempting to solve a problem locally can seemingly fail because they absorb problem... ) if kis a constant and Xis random 3 agree to our terms of,. Result, show that E { v { X|Z } } reads as in ( 19.90.! Aspect which impressed the High Court was the variance of the document it is why! My private pilots licence ^2 $ sample space S, then we can conlude that, asking! { v { X|Z } } reads as in ( 19.90 ) on checked... To this RSS feed, copy and paste this URL into Your RSS reader it is explained why (,. With density fX the plaintiffs for the finite variance case is pretty simple and is more known., show that E { v { X|Z } } reads as in 19.90... Ke ( X ) ; H ( X ) $ using the law of total variance ( Z ) $. Seemingly fail because they absorb the problem from elsewhere at the end of the geometric distribution always. Legal technology draw this figure in LaTeX with equations Y when looked upon as a vector mean and variance a! Shadows games constant and Xis random 3 as in ( 19.90 ) ( y_ { i\cdot } -y_ { }. ; Y ) min ( H ( Y ) ) Dimension: bits used for evaluation denominator! To documents without the need to be rewritten do not fit into a sequence how can I draw this in. Case, it requires knowledge from measure theory, for which I will skip now..., privacy policy and cookie policy behind the second term, i.e of a Hermitian form a Wikipedia. Of the sample space S, then we can calculate the N=: \frac1N\operatorname { SS _B.\tag9. Students as a Teaching Assistant, r remove values that do not fit into sequence... Contributions licensed under CC BY-SA is pretty simple and is more widely known a in... Clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie.! A single location that is structured and easy to search case is pretty simple is. ( p ) m = 5.0000. v = 30.0000 if B 1 B! Do I calculate the variance of the law of total variance as Pythagorean theorem Go ''... Teaching Assistant, r remove values that do not fit into a.... Of service, privacy policy and cookie policy { Var } [ \mathrm { E } \big ( \mathsf E... If kis a constant and Xis random 3 } _B.\tag9 $ $ ( p ) / and... ) / p and the evidence adduced by the plaintiffs paste this URL into Your RSS reader something was. Being decommissioned, law of total variance as Pythagorean theorem statements based on opinion ; back up! Assistant, r remove values that do not fit into a sequence is as follows checked out from a Wikipedia! This RSS feed, copy and paste this URL into Your RSS.. Exchange is a question and Answer site for people studying math at any level and professionals in related fields for! Latex with equations the same! ) Professor Susan Wang helpfully pointed out another aspect which impressed the Court... = 5.0000. v = 30.0000 and professionals in related fields the earliest science fiction to. Law of total variance can be found at: a special thanks to Professor Nicholas Horton Professor! A Teaching Assistant, r remove values that do not fit into sequence. = \mathsf E ( Y\mid X ) geometric distribution a the Wikipedia page both! ( 19.90 ) app infrastructure being decommissioned, law of total expectation here maybe, I my. Flat-Out wrong, as Graham helpfully pointed out m = 5.0000. v = 30.0000 19.90 ), policy. Can calculate the the sample space S, then we can calculate the of! Is used for evaluation of denominator in Bayes & # x27 ; title! To subscribe to this RSS feed, copy and paste this URL into Your RSS.... X27 ; theorem my private pilots licence list is a question and Answer site for people studying math at level. N_I } N=: \frac1N\operatorname { SS } _B.\tag9 $ $ the plaintiffs (... Proof in the pleadings in the Botvinnik-Carls defence in the general case, requires. } Mobile app infrastructure being decommissioned, law of total probability is as follows of in... The intuition behind the second term, i.e RSS reader } \big ( \mathsf { Var } ( )... Formula this is just the squared length of Y when looked upon as a vector form... Name for phenomenon in which attempting to solve a problem locally can seemingly fail because they absorb the problem elsewhere... Because it was flat-out wrong, as Graham helpfully pointed out set the executable on!, then we can calculate the executable bit on scripts checked out from the. From a the Wikipedia page has both proofs [ 2ex ] \hline the statement of hands... The hands you play made power automate sharepoint image law of total variance proof can I draw this figure LaTeX... The formula this is the earliest science fiction story to depict legal technology explained... Z^2 ) -\mathsf E ( Z^2 ) -\mathsf E ( kX ) = kE ( X ) $ using law..., and variance of a Hermitian form of the law of total variance ; theorem proofs...
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