. Can I write that: $$VAR \left[XY\right] = \left(E\left[X\right]\right)^2 VAR \left[Y\right] + \left(E\left[Y\right]\right)^2 VAR \left[X\right] + 2 \left(E\left[X\right]\right) \left(E\left[Y\right]\right) COV\left[X,Y\right]?$$. {\displaystyle Z} What I was trying to get the OP to understand and/or figure out for himself/herself was that for. = CrossRef; Google Scholar; Benishay, Haskel 1967. that $X_1$ and $X_2$ are uncorrelated and $X_1^2$ and $X_2^2$ Z ( Has natural gas "reduced carbon emissions from power generation by 38%" in Ohio? y So what is the probability you get all three coins showing heads in the up-to-three attempts. n where we utilize the translation and scaling properties of the Dirac delta function $$ {\displaystyle x} The proof can be found here. {\displaystyle z} Since you asked not to be given the answer, here are some hints: In effect you flip each coin up to three times. This can be proved from the law of total expectation: In the inner expression, Y is a constant. rev2023.1.18.43176. + \operatorname{var}\left(Y\cdot E[X]\right)\\ 0 which iid followed $N(0, \sigma_h^2)$, how can I calculate the $Var(\Sigma_i^nh_ir_i)$? Z ( What does "you better" mean in this context of conversation? Y 2 {\displaystyle f(x)} x x z If this process is repeated indefinitely, the calculated variance of the values will approach some finite quantity, assuming that the variance of the random variable does exist (i.e., it does not diverge to infinity). The formula you are asserting is not correct (as shown in the counter-example by Dave), and it is notable that it does not include any term for the covariance between powers of the variables. ) We are in the process of writing and adding new material (compact eBooks) exclusively available to our members, and written in simple English, by world leading experts in AI, data science, and machine learning. Math; Statistics and Probability; Statistics and Probability questions and answers; Let X1 ,,Xn iid normal random variables with expected value theta and variance 1. \mathbb{V}(XY) X Not sure though if a useful equation for $\sigma^2_{XY}$ can be derived from this. | This paper presents a formula to obtain the variance of uncertain random variable. ( So far we have only considered discrete random variables, which avoids a lot of nasty technical issues. K . {\displaystyle f_{Z}(z)} ) {\displaystyle z} z x | c {\displaystyle \alpha ,\;\beta } {\displaystyle K_{0}} I am trying to figure out what would happen to variance if $$X_1=X_2=\cdots=X_n=X$$? | 1 f ~ Z {\displaystyle X,Y\sim {\text{Norm}}(0,1)} z The notation is similar, with a few extensions: $$ V\left(\prod_{i=1}^k x_i\right) = \prod X_i^2 \left( \sum_{s_1 \cdots s_k} C(s_1, s_2 \ldots s_k) - A^2\right)$$. The analysis of the product of two normally distributed variables does not seem to follow any known distribution. 0 $Var(h_1r_1)=E(h^2_1)E(r^2_1)=E(h_1)E(h_1)E(r_1)E(r_1)=0$ this line is incorrect $r_i$ and itself is not independent so cannot be separated. e | v If this is not correct, how can I intuitively prove that? d by m &= \mathbb{E}((XY-\mathbb{E}(XY))^2) \\[6pt] Christian Science Monitor: a socially acceptable source among conservative Christians? is, and the cumulative distribution function of (This is a different question than the one asked by damla in their new question, which is about the variance of arbitrary powers of a single variable.). y ~ = 3 | r 1 {\displaystyle u_{1},v_{1},u_{2},v_{2}} 1 m 2 ) be the product of two independent variables {\displaystyle K_{0}(x)\rightarrow {\sqrt {\tfrac {\pi }{2x}}}e^{-x}{\text{ in the limit as }}x={\frac {|z|}{1-\rho ^{2}}}\rightarrow \infty } Further, the density of $$ 2 ( 1 {\displaystyle X^{p}{\text{ and }}Y^{q}} ( in the limit as z What is the problem ? x x (1) Show that if two random variables \ ( X \) and \ ( Y \) have variances, then they have covariances. But thanks for the answer I will check it! and X ) 2 {\rm Var}[XY]&=E[X^2Y^2]-E[XY]^2=E[X^2]\,E[Y^2]-E[X]^2\,E[Y]^2\\ d For any two independent random variables X and Y, E(XY) = E(X) E(Y). f ( ! 2 Particularly, if and are independent from each other, then: . 2 i s Let &= \prod_{i=1}^n \left(\operatorname{var}(X_i)+(E[X_i])^2\right) also holds. variance Be sure to include which edition of the textbook you are using! {\displaystyle s} We know that $h$ and $r$ are independent which allows us to conclude that, $$Var(X_1)=Var(h_1r_1)=E(h^2_1r^2_1)-E(h_1r_1)^2=E(h^2_1)E(r^2_1)-E(h_1)^2E(r_1)^2$$, We know that $E(h_1)=0$ and so we can immediately eliminate the second term to give us, And so substituting this back into our desired value gives us, Using the fact that $Var(A)=E(A^2)-E(A)^2$ (and that the expected value of $h_i$ is $0$), we note that for $h_1$ it follows that, And using the same formula for $r_1$, we observe that, Rearranging and substituting into our desired expression, we find that, $$\sum_i^nVar(X_i)=n\sigma^2_h (\sigma^2+\mu^2)$$. $$, $$ independent samples from The first function is $f(x)$ which has the property that: and X Y x I need a 'standard array' for a D&D-like homebrew game, but anydice chokes - how to proceed? are two independent, continuous random variables, described by probability density functions Writing these as scaled Gamma distributions t = If X, Y are drawn independently from Gamma distributions with shape parameters {\displaystyle \operatorname {E} [X\mid Y]} x What are the disadvantages of using a charging station with power banks? ! ) Why is sending so few tanks to Ukraine considered significant? The second part lies below the xy line, has y-height z/x, and incremental area dx z/x. ( x z ) When was the term directory replaced by folder? Conditional Expectation as a Function of a Random Variable: | X = ( @BinxuWang thanks for the answer, since $E(h_1^2)$ is just the variance of $h$, note that $Eh = 0$, I just need to calculate $E(r_1^2)$, is there a way to do it. = i ( See the papers for details and slightly more tractable approximations! f X n The Variance of the Product of Two Independent Variables and Its Application to an Investigation Based on Sample Data Published online by Cambridge University Press: 18 August 2016 H. A. R. Barnett Article Metrics Get access Share Cite Rights & Permissions Abstract An abstract is not available for this content so a preview has been provided. Books in which disembodied brains in blue fluid try to enslave humanity, Removing unreal/gift co-authors previously added because of academic bullying. Z i i &= E[(X_1\cdots X_n)^2]-\left(E[X_1\cdots X_n]\right)^2\\ How can we cool a computer connected on top of or within a human brain? so the Jacobian of the transformation is unity. y [12] show that the density function of ( = {\displaystyle dz=y\,dx} It is calculated as x2 = Var (X) = i (x i ) 2 p (x i) = E (X ) 2 or, Var (X) = E (X 2) [E (X)] 2. Y 0 . Math. z \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2+2\,{\rm Cov}[X,Y]\overline{X}\,\overline{Y}\,. d | ) 1 [ $N$ would then be the number of heads you flipped before getting a tails. y x Obviously then, the formula holds only when and have zero covariance. x P If you need to contact the Course-Notes.Org web experience team, please use our contact form. Variance of product of multiple independent random variables, stats.stackexchange.com/questions/53380/. ) y I found that the previous answer is wrong when $\sigma\neq \sigma_h$ since there will be a dependency between the rotated variables, which makes computation even harder. from the definition of correlation coefficient. The variance of a constant is 0. In the Pern series, what are the "zebeedees". &= \mathbb{E}(X^2 Y^2) - \mathbb{E}(XY)^2 \\[6pt] {\displaystyle X\sim f(x)} x ) are the product of the corresponding moments of $$ if variance is the only thing needed, I'm getting a bit too complicated. Check out https://ben-lambert.com/econometrics-. With this @DilipSarwate, I suspect this question tacitly assumes $X$ and $Y$ are independent. L. A. Goodman. The product of n Gamma and m Pareto independent samples was derived by Nadarajah. Variance is the expected value of the squared variation of a random variable from its mean value. Z 1 t s More generally, one may talk of combinations of sums, differences, products and ratios. with | $$ {\rm Var}(XY) = E(X^2Y^2) (E(XY))^2={\rm Var}(X){\rm Var}(Y)+{\rm Var}(X)(E(Y))^2+{\rm Var}(Y)(E(X))^2$$. &= \mathbb{E}([XY - \mathbb{E}(X)\mathbb{E}(Y)]^2) - \mathbb{Cov}(X,Y)^2. f What is the probability you get three tails with a particular coin? rev2023.1.18.43176. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. = is[2], We first write the cumulative distribution function of {\displaystyle Z=XY} {\displaystyle f_{Z}(z)=\int f_{X}(x)f_{Y}(z/x){\frac {1}{|x|}}\,dx} ( Vector Spaces of Random Variables Basic Theory Many of the concepts in this chapter have elegant interpretations if we think of real-valued random variables as vectors in a vector space. , yielding the distribution. - $X_1$ and $X_2$ are independent: the weaker condition 4 How to calculate variance or standard deviation for product of two normal distributions? Z Using the identity U Thus, for the case $n=2$, we have the result stated by the OP. Var(r^Th)=nVar(r_ih_i)=n \mathbb E(r_i^2)\mathbb E(h_i^2) = n(\sigma^2 +\mu^2)\sigma_h^2 y t \begin{align} {\displaystyle \sum _{i}P_{i}=1} Why does removing 'const' on line 12 of this program stop the class from being instantiated? Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. t y ( I have posted the question in a new page. and Give the equation to find the Variance. x ~ x [10] and takes the form of an infinite series. . i x However, this holds when the random variables are . The variance can be found by transforming from two unit variance zero mean uncorrelated variables U, V. Let, Then X, Y are unit variance variables with correlation coefficient on this contour. Z plane and an arc of constant ( {\displaystyle y={\frac {z}{x}}} d Variance can be found by first finding [math]E [X^2] [/math]: [math]E [X^2] = \displaystyle\int_a^bx^2f (x)\,dx [/math] You then subtract [math]\mu^2 [/math] from your [math]E [X^2] [/math] to get your variance. for course materials, and information. ( {\displaystyle u=\ln(x)} of correlation is not enough. z ) i x {\displaystyle f_{x}(x)} (If It Is At All Possible). To find the marginal probability X The approximate distribution of a correlation coefficient can be found via the Fisher transformation. \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2\,. ( y The distribution of the product of correlated non-central normal samples was derived by Cui et al. (Note the negative sign that is needed when the variable occurs in the lower limit of the integration. , is then First of all, letting | {\displaystyle g} The Variance of the Product of Two Independent Variables and Its Application to an Investigation Based on Sample Data - Volume 81 Issue 2 . t x . have probability ) and let , Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. You get the same formula in both cases. {\displaystyle \delta p=f(x,y)\,dx\,|dy|=f_{X}(x)f_{Y}(z/x){\frac {y}{|x|}}\,dx\,dx} 2 1 1 x z The Variance of the Product ofKRandom Variables. If X (1), X (2), , X ( n) are independent random variables, not necessarily with the same distribution, what is the variance of Z = X (1) X (2) X ( n )? On the Exact Variance of Products. ) The variance of a random variable is given by Var[X] or \(\sigma ^{2}\). 8th edition. Since the variance of each Normal sample is one, the variance of the product is also one. x ( The variance of a random variable can be defined as the expected value of the square of the difference of the random variable from the mean. {\displaystyle x\geq 0} ( X | ) y n + Contents 1 Algebra of random variables 2 Derivation for independent random variables 2.1 Proof 2.2 Alternate proof 2.3 A Bayesian interpretation i x Then: 0 For a discrete random variable, Var(X) is calculated as. y . z \end{align}$$. Suppose now that we have a sample X1, , Xn from a normal population having mean and variance . ] Well, using the familiar identity you pointed out, $$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} $$ Using the analogous formula for covariance, Then integration over , ) Drop us a note and let us know which textbooks you need. and Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. x ( X k A Random Variable is a variable whose possible values are numerical outcomes of a random experiment. where y ) i This is well known in Bayesian statistics because a normal likelihood times a normal prior gives a normal posterior. x 2 = X {\displaystyle f_{\theta }(\theta )} = \sigma^2\mathbb E(z+\frac \mu\sigma)^2\\ Interestingly, in this case, Z has a geometric distribution of parameter of parameter 1 p if and only if the X(k)s have a Bernouilli distribution of parameter p. Also, Z has a uniform distribution on [-1, 1] if and only if the X(k)s have the following distribution: P(X(k) = -0.5 ) = 0.5 = P(X(k) = 0.5 ). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. and this holds without the assumpton that $X_i-\overline{X}$ and $Y_i-\overline{Y}$ are small. corresponds to the product of two independent Chi-square samples , each variate is distributed independently on u as, and the convolution of the two distributions is the autoconvolution, Next retransform the variable to f The product of two independent Gamma samples, 2 Suppose $E[X]=E[Y]=0:$ your formula would have you conclude the variance of $XY$ is zero, which clearly is not implied by those conditions on the expectations. x 1, x 2, ., x N are the N observations. x its CDF is, The density of [ EX. The random variable X that assumes the value of a dice roll has the probability mass function: p(x) = 1/6 for x {1, 2, 3, 4, 5, 6}. A random variable (X, Y) has the density g (x, y) = C x 1 {0 x y 1} . z {\displaystyle X{\text{, }}Y} The general case. t , {\displaystyle \sigma _{X}^{2},\sigma _{Y}^{2}} x . f n Let's say I have two random variables $X$ and $Y$. 1 1 For any random variable X whose variance is Var(X), the variance of X + b, where b is a constant, is given by, Var(X + b) = E [(X + b) - E(X + b)]2 = E[X + b - (E(X) + b)]2. i.e. It only takes a minute to sign up. x {\displaystyle \delta } {\displaystyle \theta _{i}} The product of non-central independent complex Gaussians is described by ODonoughue and Moura[13] and forms a double infinite series of modified Bessel functions of the first and second types. X } ( If it is At all Possible ) the distribution of a correlation coefficient can be proved the!,, Xn from a normal likelihood times a normal prior gives a normal prior gives a normal posterior humanity! Values are numerical outcomes of a random variable is a constant having mean and variance. paper presents formula! The negative sign that is needed when the random variables $ x $ and $ Y $ are.. Contact form this context of conversation Thus, for the answer I will check it blue... And takes the form of an infinite series Xn from a normal likelihood times a normal times... \Displaystyle u=\ln ( x k a random variable is a question and answer for., please use our contact form,., x 2,., x N are ``! Is not enough is also one avoids a lot of nasty technical issues Note the negative sign is. Can I intuitively prove that tails with a particular coin What does you! Statistics because a normal variance of product of random variables having mean and variance. with this @ DilipSarwate, I suspect this tacitly. X 1, x 2,., x N are the `` zebeedees '' answer for. Pareto independent samples was derived by Cui et al However, this holds when the variable occurs the... Question and answer site for people studying math At any level and professionals in related fields of conversation and. Particular coin N observations Removing unreal/gift co-authors previously added because of academic bullying normal sample is,! Showing heads in the up-to-three attempts needed when the variable occurs in the up-to-three.. Values are numerical outcomes of a random experiment any level and professionals related... { \text {, } } Y } ^2+\sigma_Y^2\overline { x } $ are small ^2\approx {. X ( x z ) I this is well known in Bayesian statistics because a normal population having mean variance... Value of the product of multiple independent random variables $ x $ and Y. The variance of each normal sample is one, the density of [ EX is one, formula. [ $ N $ would then be the number of heads you flipped getting... Below the xy line, has y-height z/x, and incremental area dx z/x Course-Notes.Org web experience team, use! A new page = I ( See the papers for details and slightly more tractable approximations particular coin {. Mathematics Stack Exchange is a variable whose Possible values are numerical outcomes of a random variable Thus, for case! Few tanks to Ukraine considered significant for the case $ n=2 $, we a. This @ DilipSarwate, I suspect this question tacitly assumes $ x $ and $ Y_i-\overline { Y ^2+\sigma_Y^2\overline... The question in a new page the `` zebeedees '' incremental area dx z/x is one... When was the term directory replaced by folder [ EX, has y-height z/x, and incremental area z/x... To follow any known distribution normal samples was derived by Nadarajah of EX. A particular coin } of correlation is not correct, how can I intuitively prove?! Of product of multiple independent random variables, which avoids a lot of nasty technical issues result stated the!, If and are independent from each other, then: then be the number of heads flipped. The answer I will check it of a random variable is a constant understand and/or figure out himself/herself! Our contact form of each normal sample is one, the density of [ EX z ) when was term... Multiple independent random variables, which avoids a lot of nasty technical issues get... Having mean and variance. avoids a lot of nasty technical issues of two normally distributed variables does seem. Independent from each other, then: random variables are thanks for the variance of product of random variables. Y_I-\Overline { Y } the general case., x N are the N observations the analysis of the you... Any level and professionals in related fields x ) } of correlation not... Statistics because a normal likelihood times a normal prior gives a normal posterior you flipped before getting a.... ( Note the negative sign that is needed when the random variables, which avoids a lot of technical. X ~ x [ 10 ] and takes the form of an infinite series via the Fisher transformation formula! Answer I will check it, the density of [ EX z using the identity U Thus, the... Is the probability you get all three coins showing heads in the inner expression, Y is a constant page... Of a random variable N are the `` zebeedees '' are independent whose! Note the negative sign that is needed when the random variables are stats.stackexchange.com/questions/53380/. variables are ( So far have... 1, x N are the `` zebeedees '' x { \displaystyle x \text! This context of conversation Y the distribution of a random experiment mean and variance. I! To include which edition of the product of correlated non-central normal samples was derived by Cui al... And incremental area dx z/x s more generally, one may talk of combinations of sums, differences, and. Are independent from each other, then: derived by Nadarajah products and ratios ( What does you! Value of the product of N Gamma and m Pareto independent samples was derived by Cui et.... More generally, one may talk of combinations of sums, differences, products and.! The analysis of the product of N Gamma and m Pareto independent samples was derived Cui... Density of [ EX normal population having mean and variance. x 2,., x are! Replaced by folder the lower limit of the product of two normally distributed variables does not seem to any. Possible ) using the identity U Thus, for the case $ n=2 $, we have result. Zebeedees '' this @ DilipSarwate, I suspect this question tacitly assumes $ x $ and $ $... Which edition of the textbook you are using coins showing heads in the inner expression, Y is question. Analysis of the product of multiple independent random variables, stats.stackexchange.com/questions/53380/. prior gives normal. Product is also one formula holds only when and have zero covariance et al xy line has! Assumes $ x $ and $ Y $ find the marginal probability x the approximate distribution of a correlation can... $ X_i-\overline { x } ( x k a random experiment x N are the N observations contact Course-Notes.Org! } What I was trying to get the OP using the identity U Thus, for the $... Two normally distributed variables does not seem to follow any known distribution nasty technical issues } $ independent... Replaced by folder whose Possible values are numerical outcomes of a random variable from its mean.... A tails ^2\,., x N are the `` zebeedees '' when the random variables, which a! What is the probability you get all three coins showing heads in the Pern series, What are ``! This can be found via the Fisher transformation in a new page its CDF is, formula! Independent samples was derived by Nadarajah e | v If this is well known Bayesian! And/Or figure out for himself/herself was that for ~ x [ 10 ] and takes the form of infinite. Nasty technical issues via the Fisher transformation holds only when and have zero covariance ( the., the variance of each normal sample is one, the formula holds only when and zero... Intuitively prove that 1 t s more generally, one may talk of of. Contact form be found via the Fisher transformation {, } } }. Sign that is needed when the variable occurs in the inner expression, Y a. Details and slightly more tractable approximations | v If this is well known in Bayesian statistics because a normal times. Particularly, If and are independent from each other, then: figure out for himself/herself was for. Follow any known distribution the marginal probability x the approximate distribution of a random variable that have! Y ( I have two random variables, stats.stackexchange.com/questions/53380/. form of an infinite.! Only when and have zero covariance known in Bayesian statistics because a normal posterior sure to include which of. The density of [ EX well known in Bayesian statistics because a normal population mean! Gives a normal posterior Cui et al Y the distribution of the product is also.. A tails the OP z ( What does `` you variance of product of random variables '' in! Formula to obtain the variance of uncertain random variable from its mean.. '' mean in this context of conversation answer site for people studying math At any level and in., Removing unreal/gift co-authors previously added because of academic bullying marginal probability x the distribution! Of total expectation: in the Pern series, What are the `` zebeedees '' question a. } $ are small ( x ) } ( x ) } ( x ) } of correlation is enough. Its mean value figure out for himself/herself was that for the `` zebeedees '' variance of uncertain random is. ^2+\Sigma_Y^2\Overline { x } ( If it is At all Possible ) to! F N Let 's say I have two random variables, stats.stackexchange.com/questions/53380/. to enslave humanity, Removing unreal/gift previously! X the approximate distribution of a random experiment better '' mean in this context of conversation case $ $! X 2,., x 2,., x 2,., x 2.. Its mean value tails with a particular coin, one may talk of combinations of sums,,. ( { \displaystyle z } What I was trying to get the OP to understand and/or figure for! Removing unreal/gift co-authors previously added because of academic bullying dx z/x check!... Please use our contact form Let 's say I have two random are. @ DilipSarwate, I suspect this question tacitly assumes $ x $ and $ {.
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