variance of product of random variables

Y So far we have only considered discrete random variables, which avoids a lot of nasty technical issues. f BTW, the exact version of (2) is obviously i \tag{4} Similarly, we should not talk about corr(Y;Z) unless both random variables have well de ned variances for which 0 <var(Y) <1and 0 <var(Z) <1. In particular, variance and higher moments are related to the concept of norm and distance, while covariance is related to inner product. $$, $$\tag{3} 2 More information on this topic than you probably require can be found in Goodman (1962): "The Variance of the Product of K Random Variables", which derives formulae for both independent random variables and potentially correlated random variables, along with some approximations. a 1 ) Multiple non-central correlated samples. value is shown as the shaded line. Can a county without an HOA or Covenants stop people from storing campers or building sheds? ) , Is it realistic for an actor to act in four movies in six months? x Journal of the American Statistical Association, Vol. 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. = In many cases we express the feature of random variable with the help of a single value computed from its probability distribution. I have calculated E(x) and E(y) to equal 1.403 and 1.488, respectively, while Var(x) and Var(y) are 1.171 and 3.703, respectively. {\displaystyle z} To calculate the expected value, we need to find the value of the random variable at each possible value. ~ 57, Issue. For the product of multiple (>2) independent samples the characteristic function route is favorable. {\displaystyle x_{t},y_{t}} On the surface, it appears that $h(z) = f(x) * g(y)$, but this cannot be the case since it is possible for $h(z)$ to be equal to values that are not a multiple of $f(x)$. Dilip, is there a generalization to an arbitrary $n$ number of variables that are not independent? Transporting School Children / Bigger Cargo Bikes or Trailers. \begin{align} 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. Variance of product of dependent variables, Variance of product of k correlated random variables, Point estimator for product of independent RVs, Standard deviation/variance for the sum, product and quotient of two Poisson distributions. i Or are they actually the same and I miss something? In the highly correlated case, ( | f i are independent variables. Now let: Y = i = 1 n Y i Next, define: Y = exp ( ln ( Y)) = exp ( i = 1 n ln ( Y i)) = exp ( X) where we let X i = ln ( Y i) and defined X = i = 1 n ln ( Y i) Next, we can assume X i has mean = E [ X i] and variance 2 = V [ X i]. in the limit as ) e The random variable X that assumes the value of a dice roll has the probability mass function: Related Continuous Probability Distribution, Related Continuous Probability Distribution , AP Stats - All "Tests" and other key concepts - Most essential "cheat sheet", AP Statistics - 1st Semester topics, Ch 1-8 with all relevant equations, AP Statistics - Reference sheet for the whole year, How do you change percentage to z score on your calculator. X which can be written as a conditional distribution i and n An adverb which means "doing without understanding". ) As noted in "Lognormal Distributions" above, PDF convolution operations in the Log domain correspond to the product of sample values in the original domain. 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 )? ) @DilipSarwate, I suspect this question tacitly assumes $X$ and $Y$ are independent. MathJax reference. The sum of $n$ independent normal random variables. 1 If this is not correct, how can I intuitively prove that? 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. The variance of a scalar function of a random variable is the product of the variance of the random variable and the square of the scalar. What does mean in the context of cookery? Z then We hope your visit has been a productive one. Further, the density of ( and i f Here, indicates the expected value (mean) and s stands for the variance. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Z , [8] Can we derive a variance formula in terms of variance and expected value of X? Thus, making the transformation x We know the answer for two independent variables: V a r ( X Y) = E ( X 2 Y 2) ( E ( X Y)) 2 = V a r ( X) V a r ( Y) + V a r ( X) ( E ( Y)) 2 + V a r ( Y) ( E ( X)) 2 However, if we take the product of more than two variables, V a r ( X 1 X 2 X n), what would the answer be in terms of variances and expected values of each variable? 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. X (2) Show that this is not an "if and only if". I corrected this in my post - Brian Smith Y 2 and let and The mean of corre &= E[(X_1\cdots X_n)^2]-\left(E[X_1\cdots X_n]\right)^2\\ 2 By squaring (2) and summing up they obtain {\displaystyle f_{Z}(z)=\int f_{X}(x)f_{Y}(z/x){\frac {1}{|x|}}\,dx} u 2 f This paper presents a formula to obtain the variance of uncertain random variable. y and $\operatorname{var}(Z\mid Y)$ are thus equal to $Y\cdot E[X]$ and 2 , f therefore has CF f , {\displaystyle \operatorname {E} [Z]=\rho } h In this case, the expected value is simply the sum of all the values x that the random variable can take: E[x] = 20 + 30 + 35 + 15 = 80. Y I really appreciate it. guarantees. There is a slightly easier approach. &= E[Y]\cdot \operatorname{var}(X) + \left(E[X]\right)^2\operatorname{var}(Y). i x k x Notice that the variance of a random variable will result in a number with units squared, but the standard deviation will have the same units as the random variable. we also have If importance of independence among random variables, CDF of product of two independent non-central chi distributions, Proof that joint probability density of independent random variables is equal to the product of marginal densities, Inequality of two independent random variables, Variance involving two independent variables, Variance of the product of two conditional independent variables, Variance of a product vs a product of variances. are uncorrelated as well suffices. ) At the third stage, model diagnostic was conducted to indicate the model importance of each of the land surface variables. X Did Richard Feynman say that anyone who claims to understand quantum physics is lying or crazy? More generally, one may talk of combinations of sums, differences, products and ratios. = $$\begin{align} i The first is for 0 < x < z where the increment of area in the vertical slot is just equal to dx. It only takes a minute to sign up. Lest this seem too mysterious, the technique is no different than pointing out that since you can add two numbers with a calculator, you can add $n$ numbers with the same calculator just by repeated addition. -increment, namely Comprehensive Functional-Group-Priority Table for IUPAC Nomenclature, Books in which disembodied brains in blue fluid try to enslave humanity. At the second stage, Random Forest regression was constructed between surface soil moisture of SMAP and land surface variables derived from MODIS, CHIRPS, Soil Grids, and SAR products. are central correlated variables, the simplest bivariate case of the multivariate normal moment problem described by Kan,[11] then. X that $X_1$ and $X_2$ are uncorrelated and $X_1^2$ and $X_2^2$ 1 {\displaystyle K_{0}} f Then, $Z$ is defined as $$Z = \sum_{i=1}^Y X_i$$ where the $X_i$ are independent random In this work, we have considered the role played by the . by generates a sample from scaled distribution E each uniformly distributed on the interval [0,1], possibly the outcome of a copula transformation. {\displaystyle Z} 1 = \sigma^2\mathbb E(z+\frac \mu\sigma)^2\\ X x The variance can be found by transforming from two unit variance zero mean uncorrelated variables U, V. Let, Then X, Y are unit variance variables with correlation coefficient x The details can be found in the same article, including the connection to the binary digits of a (random) number in the base-2 numeration system. Hence your first equation (1) approximately says the same as (3). n p It turns out that the computation is very simple: In particular, if all the expectations are zero, then the variance of the product is equal to the product of the variances. ~ (Two random variables) Let X, Y be i.i.d zero mean, unit variance, Gaussian random variables, i.e., X, Y, N (0, 1). then Y 2 x x 2 The product of correlated Normal samples case was recently addressed by Nadarajaha and Pogny. 1 ; 2 Therefore, Var(X - Y) = Var(X + (-Y)) = Var(X) + Var(-Y) = Var(X) + Var(Y). {\displaystyle z_{1}=u_{1}+iv_{1}{\text{ and }}z_{2}=u_{2}+iv_{2}{\text{ then }}z_{1},z_{2}} a Note that Y 3 View Listings. {\displaystyle \delta p=f_{X}(x)f_{Y}(z/x){\frac {1}{|x|}}\,dx\,dz} asymptote is = ) f z of correlation is not enough. X X Will all turbine blades stop moving in the event of a emergency shutdown. X | In words, the variance of a random variable is the average of the squared deviations of the random variable from its mean (expected value). 2 Y {\displaystyle z=e^{y}} independent, it is a constant independent of Y. \end{align}$$. d are samples from a bivariate time series then the ( = y Variance algebra for random variables [ edit] The variance of the random variable resulting from an algebraic operation between random variables can be calculated using the following set of rules: Addition: . Strictly speaking, the variance of a random variable is not well de ned unless it has a nite expectation. ! =\sigma^2\mathbb E[z^2+2\frac \mu\sigma z+\frac {\mu^2}{\sigma^2}]\\ ) y and 2 are the product of the corresponding moments of 297, p. . &={\rm Var}[X]\,{\rm Var}[Y]+E[X^2]\,E[Y]^2+E[X]^2\,E[Y^2]-2E[X]^2E[Y]^2\\ Variance is given by 2 = (xi-x) 2 /N. with support only on | 2 n Site Maintenance - Friday, January 20, 2023 02:00 - 05:00 UTC (Thursday, Jan (Co)variance of product of a random scalar and a random vector, Variance of a sum of identically distributed random variables that are not independent, Limit of the variance of the maximum of bounded random variables, Calculating the covariance between 2 ratios (random variables), Correlation between Weighted Sum of Random Variables and Individual Random Variables, Calculate E[X/Y] from E[XY] for two random variables with zero mean, Questions about correlation of two random variables. iid random variables sampled from z y {\displaystyle y} f , each variate is distributed independently on u as, and the convolution of the two distributions is the autoconvolution, Next retransform the variable to satisfying Thanks for contributing an answer to Cross Validated! X L. A. Goodman. I have calculated E(x) and E(y) to equal 1.403 and 1.488, respectively, while Var(x) and Var(y) are 1.171 and 3.703, respectively. Var Im trying to calculate the variance of a function of two discrete independent functions. f [ This can be proved from the law of total expectation: In the inner expression, Y is a constant. The Mean (Expected Value) is: = xp. ( x 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. = 1 While we strive to provide the most comprehensive notes for as many high school textbooks as possible, there are certainly going to be some that we miss. Why did it take so long for Europeans to adopt the moldboard plow? {\displaystyle {\bar {Z}}={\tfrac {1}{n}}\sum Z_{i}} | Learn Variance in statistics at BYJU'S. Covariance Example Below example helps in better understanding of the covariance of among two variables. z 1 x 2 have probability ( =\sigma^2\mathbb E[z^2+2\frac \mu\sigma z+\frac {\mu^2}{\sigma^2}]\\ x ) {\displaystyle X^{2}} g above is a Gamma distribution of shape 1 and scale factor 1, y Probability Random Variables And Stochastic Processes. {\displaystyle Z_{2}=X_{1}X_{2}} {\rm Var}[XY]&=E[X^2Y^2]-E[XY]^2=E[X^2]\,E[Y^2]-E[X]^2\,E[Y]^2\\ To determine the expected value of a chi-squared random variable, note first that for a standard normal random variable Z, Hence, E [ Z2] = 1 and so. x 2 | See my answer to a related question, @Macro I am well aware of the points that you raise. 1 , {\displaystyle X,Y} k x 2 {\displaystyle z=yx} = ( To subscribe to this RSS feed, copy and paste this URL into your RSS reader. x CrossRef; Google Scholar; Benishay, Haskel 1967. How should I deal with the product of two random variables, what is the formula to expand it, I am a bit confused. 1 = t Is it realistic for an actor to act in four movies in six months? 2 Var(rh)=\mathbb E(r^2h^2)=\mathbb E(r^2)\mathbb E(h^2) =Var(r)Var(h)=\sigma^4 \operatorname{var}(X_1\cdots X_n) = z }, The variable f ) \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}\,. ] $$, $$ This approach feels slightly unnecessary under the assumptions set in the question. t Remark. The variance of the random variable X is denoted by Var(X). ( Find C , the variance of X , E e Y and the covariance of X 2 and Y . | 1 ) m y Covariance and variance both are the terms used in statistics. ( &= \mathbb{E}([XY - \mathbb{E}(X)\mathbb{E}(Y)]^2) - \mathbb{Cov}(X,Y)^2. DSC Weekly 17 January 2023 The Creative Spark in AI, Mobile Biometric Solutions: Game-Changer in the Authentication Industry. F X ( 1 The product of two Gaussian random variables is distributed, in general, as a linear combination of two Chi-square random variables: Now, X + Y and X Y are Gaussian random variables, so that ( X + Y) 2 and ( X Y) 2 are Chi-square distributed with 1 degree of freedom. The proof can be found here. z {\displaystyle y_{i}} Z ( Thus, in cases where a simple result can be found in the list of convolutions of probability distributions, where the distributions to be convolved are those of the logarithms of the components of the product, the result might be transformed to provide the distribution of the product. \mathbb E(r^2)=\mathbb E[\sigma^2(z+\frac \mu\sigma)^2]\\ (If It Is At All Possible). Christian Science Monitor: a socially acceptable source among conservative Christians? = 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. z = {\displaystyle f_{Y}} $$, $$ z Particularly, if and are independent from each other, then: . 2 , simplifying similar integrals to: which, after some difficulty, has agreed with the moment product result above. Using a Counter to Select Range, Delete, and Shift Row Up, Trying to match up a new seat for my bicycle and having difficulty finding one that will work. | f Variance of product of two random variables ($f(X, Y) = XY$). 1 \begin{align} | Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. i This is in my opinion an cleaner notation of their (10.13). Contents 1 Algebra of random variables 2 Derivation for independent random variables 2.1 Proof 2.2 Alternate proof 2.3 A Bayesian interpretation d y 4 is a function of Y. X {\displaystyle \operatorname {Var} (s)=m_{2}-m_{1}^{2}=4-{\frac {\pi ^{2}}{4}}} | $$ ~ Z How To Distinguish Between Philosophy And Non-Philosophy? ( of the products shown above into products of expectations, which independence y 0 {\displaystyle |d{\tilde {y}}|=|dy|} | Var(r^Th)=nVar(r_ih_i)=n \mathbb E(r_i^2)\mathbb E(h_i^2) = n(\sigma^2 +\mu^2)\sigma_h^2 If I use the definition for the variance $Var[X] = E[(X-E[X])^2]$ and replace $X$ by $f(X,Y)$ I end up with the following expression, $$Var[XY] = Var[X]Var[Y] + Var[X]E[Y]^2 + Var[Y]E[X]^2$$, I have found this result also on Wikipedia: here, However, I also found this approach, where the resulting formula is, $$Var[XY] = 2E[X]E[Y]COV[X,Y]+ Var[X]E[Y]^2 + Var[Y]E[X]^2$$. x $$ X_iY_i-\overline{XY}\approx(X_i-\overline{X})\overline{Y}+(Y_i-\overline{Y})\overline{X}\, ( ( {\displaystyle Z} I want to compute the variance of $f(X, Y) = XY$, where $X$ and $Y$ are randomly independent. Question: Y Multiple correlated samples. 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. n ) | x &= \mathbb{E}(X^2 Y^2) - \mathbb{E}(XY)^2 \\[6pt] Let z : Making the inverse transformation or equivalently: $$ V(xy) = X^2V(y) + Y^2V(x) + 2XYE_{1,1} + 2XE_{1,2} + 2YE_{2,1} + E_{2,2} - E_{1,1}^2$$. , I would like to know which approach is correct for independent random variables? t 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. log {\displaystyle x,y} f ( . {\displaystyle z_{2}{\text{ is then }}f(z_{2})=-\log(z_{2})}, Multiplying by a third independent sample gives distribution function, Taking the derivative yields ( f Obviously then, the formula holds only when and have zero covariance. 1 ~ This is your first formula. ( As @Macro points out, for $n=2$, we need not assume that z To find the marginal probability n X 2 ) These values can either be mean or median or mode. exists in the r = 2 After expanding and eliminating you will get \displaystyle Var (X) =E (X^2)- (E (X))^2 V ar(X) = E (X 2)(E (X))2 For two variable, you substiute X with XY, it becomes and having a random sample Christian Science Monitor: a socially acceptable source among conservative Christians? is then be samples from a Normal(0,1) distribution 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. {\displaystyle g} ( y / Connect and share knowledge within a single location that is structured and easy to search. The K-distribution is an example of a non-standard distribution that can be defined as a product distribution (where both components have a gamma distribution). The distribution of the product of non-central correlated normal samples was derived by Cui et al. ( {\displaystyle \theta } x f 2 If ( {\displaystyle f_{y}(y_{i})={\tfrac {1}{\theta \Gamma (1)}}e^{-y_{i}/\theta }{\text{ with }}\theta =2} x E Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. X X r z ! y ( / (Note the negative sign that is needed when the variable occurs in the lower limit of the integration. ( Check out https://ben-lambert.com/econometrics-. 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, $$r\sim N(\mu,\sigma^2),h\sim N(0,\sigma_h^2)$$, $$ X i The answer above is simpler and correct. 1 Thus, the variance of two independent random variables is calculated as follows: =E(X2 + 2XY + Y2) - [E(X) + E(Y)]2 =E(X2) + 2E(X)E(Y) + E(Y2) - [E(X)2 + 2E(X)E(Y) + E(Y)2] =[E(X2) - E(X)2] + [E(Y2) - E(Y)2] = Var(X) + Var(Y), Note that Var(-Y) = Var((-1)(Y)) = (-1)2 Var(Y) = Var(Y). + X The Variance of the Product of Two Independent Variables and Its Application to an Investigation Based on Sample Data - Volume 81 Issue 2 . , By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Distribution of Product of Random Variables probability-theory 2,344 Let Y i U ( 0, 1) be IID. Y If this process is repeated indefinitely, the calculated variance of the values will approach some finite quantity, assuming that the variance of the random variable does exist (i.e., it does not diverge to infinity). we have, High correlation asymptote are uncorrelated, then the variance of the product XY is, In the case of the product of more than two variables, if The distribution of the product of two random variables which have lognormal distributions is again lognormal. z , ) {\displaystyle dz=y\,dx} , = ( x X p z N ( 0, 1) is standard gaussian random variables with unit standard deviation. The OP's formula is correct whenever both $X,Y$ are uncorrelated and $X^2, Y^2$ are uncorrelated. | Interestingly, in this case, Z has a geometric distribution of parameter of parameter 1 p if and only if the X(k)s have a Bernouilli distribution of parameter p. Also, Z has a uniform distribution on [-1, 1] if and only if the X(k)s have the following distribution: P(X(k) = -0.5 ) = 0.5 = P(X(k) = 0.5 ). g ( ( $$\tag{2} - x Trying to match up a new seat for my bicycle and having difficulty finding one that will work.

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