variance of product of random variables

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. x z {\displaystyle h_{X}(x)} and. x The characteristic function of X is ( {\displaystyle X_{1}\cdots X_{n},\;\;n>2} In the special case in which X and Y are statistically . $$. 1 Or are they actually the same and I miss something? and variances The Mean (Expected Value) is: = xp. x x {\displaystyle x,y} Drop us a note and let us know which textbooks you need. Downloadable (with restrictions)! {\displaystyle u_{1},v_{1},u_{2},v_{2}} | f &= \mathbb{E}([XY - \mathbb{E}(X)\mathbb{E}(Y)]^2) - \mathbb{Cov}(X,Y)^2. {\displaystyle n!!} {\displaystyle X,Y\sim {\text{Norm}}(0,1)} But because Bayesian applications don't usually need to know the proportionality constant, it's a little hard to find. {\displaystyle f_{Z}(z)} h , 0 The pdf of a function can be reconstructed from its moments using the saddlepoint approximation method. x z Journal of the American Statistical Association. ( The analysis of the product of two normally distributed variables does not seem to follow any known distribution. = y This finite value is the variance of the random variable. 1 =\sigma^2+\mu^2 x ( The variance of uncertain random variable may provide a degree of the spread of the distribution around its expected value. ( = X {\displaystyle \sigma _{X}^{2},\sigma _{Y}^{2}} | be the product of two independent variables A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Let What does mean in the context of cookery? {\displaystyle \theta X\sim {\frac {1}{|\theta |}}f_{X}\left({\frac {x}{\theta }}\right)} {\displaystyle X^{p}{\text{ and }}Y^{q}} Y The variance is the standard deviation squared, and so is often denoted by {eq}\sigma^2 {/eq}. Will all turbine blades stop moving in the event of a emergency shutdown. n Similarly, the variance of the sum or difference of a set of independent random variables is simply the sum of the variances of the independent random variables in the set. If we are not too sure of the result, take a special case where $n=1,\mu=0,\sigma=\sigma_h$, then we know ( In the Pern series, what are the "zebeedees". | Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. X_iY_i-\overline{XY}\approx(X_i-\overline{X})\overline{Y}+(Y_i-\overline{Y})\overline{X}\, ( $$, $$ p are uncorrelated, then the variance of the product XY is, In the case of the product of more than two variables, if {\displaystyle W=\sum _{t=1}^{K}{\dbinom {x_{t}}{y_{t}}}{\dbinom {x_{t}}{y_{t}}}^{T}} {\displaystyle (1-it)^{-1}} ( Variance of the sum of two random variables Let and be two random variables. Z m | What does mean in the context of cookery? Hence: This is true even if X and Y are statistically dependent in which case t 0 1 [ ) ] Find C , the variance of X , E e Y and the covariance of X 2 and Y . Thus, for the case $n=2$, we have the result stated by the OP. E (X 2) = i x i2 p (x i ), and [E (X)] 2 = [ i x i p (x i )] 2 = 2. (Note the negative sign that is needed when the variable occurs in the lower limit of the integration. is, and the cumulative distribution function of If I use the definition for the variance $Var[X] = E[(X-E[X])^2]$ and replace $X$ by $f(X,Y)$ I end up with the following expression, $$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$$. Under the given conditions, $\mathbb E(h^2)=Var(h)=\sigma_h^2$. Note the non-central Chi sq distribution is the sum $k $independent, normally distributed random variables with means $\mu_i$ and unit variances. x is a Wishart matrix with K degrees of freedom. i X y are two independent random samples from different distributions, then the Mellin transform of their product is equal to the product of their Mellin transforms: If s is restricted to integer values, a simpler result is, Thus the moments of the random product To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Why did it take so long for Europeans to adopt the moldboard plow? - . Then, The variance of this distribution could be determined, in principle, by a definite integral from Gradsheyn and Ryzhik,[7], thus i is. ! appears only in the integration limits, the derivative is easily performed using the fundamental theorem of calculus and the chain rule. d X . | If I use the definition for the variance V a r [ X] = E [ ( X E [ X]) 2] and replace X by f ( X, Y) I end up with the following expression f n How can citizens assist at an aircraft crash site? Foundations Of Quantitative Finance Book Ii: Probability Spaces And Random Variables order online from Donner! ( I have calculated E(x) and E(y) to equal 1.403 and 1.488, respectively, while Var(x) and Var(y) are 1.171 and 3.703, respectively. each with two DoF. The post that the original answer is based on is this. = Z Variance is the expected value of the squared variation of a random variable from its mean value. {\displaystyle x} Coding vs Programming Whats the Difference? = The formula you are asserting is not correct (as shown in the counter-example by Dave), and it is notable that it does not include any term for the covariance between powers of the variables. I am trying to figure out what would happen to variance if $$X_1=X_2=\cdots=X_n=X$$? and all the X(k)s are independent and have the same distribution, then we have. Consider the independent random variables X N (0, 1) and Y N (0, 1). i {\displaystyle Z} {\displaystyle Z=X_{1}X_{2}} See the papers for details and slightly more tractable approximations! ) z K $$V(xy) = (XY)^2[G(y) + G(x) + 2D_{1,1} + 2D_{1,2} + 2D_{2,1} + D_{2,2} - D_{1,1}^2] $$ ! Therefore, Var(X - Y) = Var(X + (-Y)) = Var(X) + Var(-Y) = Var(X) + Var(Y). z , ) + \operatorname{var}\left(Y\cdot E[X]\right)\\ The random variables $E[Z\mid Y]$ ( Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Z that $X_1$ and $X_2$ are uncorrelated and $X_1^2$ and $X_2^2$ Z Thus, making the transformation x h = ) $$\begin{align} ) Has natural gas "reduced carbon emissions from power generation by 38%" in Ohio? \mathbb{V}(XY) z ) Why did it take so long for Europeans to adopt the moldboard plow? 2 &= E[X_1^2\cdots X_n^2]-\left(E[(X_1]\cdots E[X_n]\right)^2\\ X Var 1 which has the same form as the product distribution above. &= \mathbb{E}(([XY - \mathbb{E}(X)\mathbb{E}(Y)] - \mathbb{Cov}(X,Y))^2) \\[6pt] ( Variance: The variance of a random variable is a measurement of how spread out the data is from the mean. i n r x x $$\tag{2} {\displaystyle xy\leq z} exists in the Does the LM317 voltage regulator have a minimum current output of 1.5 A. 2 What is the problem ? s Welcome to the newly launched Education Spotlight page! {\displaystyle |d{\tilde {y}}|=|dy|} X (a) Derive the probability that X 2 + Y 2 1. Christian Science Monitor: a socially acceptable source among conservative Christians? &={\rm Var}[X]\,{\rm Var}[Y]+{\rm Var}[X]\,E[Y]^2+{\rm Var}[Y]\,E[X]^2\,. If we see enough demand, we'll do whatever we can to get those notes up on the site for you! {\rm Var}[XY]&=E[X^2Y^2]-E[XY]^2=E[X^2]\,E[Y^2]-E[X]^2\,E[Y]^2\\ Theorem 8 (Chebyshev's Theorem) Let X be a random variable, then for any k . Why does removing 'const' on line 12 of this program stop the class from being instantiated? ( \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2\,. K x I found that the previous answer is wrong when $\sigma\neq \sigma_h$ since there will be a dependency between the rotated variables, which makes computation even harder. ) 1 I used the moment generating function of normal distribution and take derivative wrt t twice and set it to zero and got it. ) It only takes a minute to sign up. each uniformly distributed on the interval [0,1], possibly the outcome of a copula transformation. ; variables with the same distribution as $X$. , nl / en; nl / en; Customer support; Login; Wish list; 0. checkout No shipping costs from 15, - Lists and tips from our own specialists Possibility of ordering without an account . = X or equivalently it is clear that What to make of Deepminds Sparrow: Is it a sparrow or a hawk? x Z Y | are central correlated variables, the simplest bivariate case of the multivariate normal moment problem described by Kan,[11] then. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product f What is required is the factoring of the expectation $$ \end{align}$$ In the Pern series, what are the "zebeedees"? . 2 The general case. i f x Can we derive a variance formula in terms of variance and expected value of X? 2 However, $XY\sim\chi^2_1$, which has a variance of $2$. / Is it also possible to do the same thing for dependent variables? with &= \mathbb{E}((XY - \mathbb{Cov}(X,Y) - \mathbb{E}(X)\mathbb{E}(Y))^2) \\[6pt] with parameters {\displaystyle \Gamma (x;k_{i},\theta _{i})={\frac {x^{k_{i}-1}e^{-x/\theta _{i}}}{\Gamma (k_{i})\theta _{i}^{k_{i}}}}} y Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. e 1 How to tell a vertex to have its normal perpendicular to the tangent of its edge? ) {\displaystyle y} &= \mathbb{E}(X^2 Y^2) - \mathbb{E}(XY)^2 \\[6pt] are y s 2 On the Exact Variance of Products. = {\displaystyle Z} 0 1 \tag{4} then, This type of result is universally true, since for bivariate independent variables f f Did Richard Feynman say that anyone who claims to understand quantum physics is lying or crazy? 1 {\displaystyle K_{0}(x)\rightarrow {\sqrt {\tfrac {\pi }{2x}}}e^{-x}{\text{ in the limit as }}x={\frac {|z|}{1-\rho ^{2}}}\rightarrow \infty } x rev2023.1.18.43176. x {\displaystyle X{\text{, }}Y} | These product distributions are somewhat comparable to the Wishart distribution. , and the distribution of Y is known. | $$ ( x Independence suffices, but ( x ~ 1 If you need to contact the Course-Notes.Org web experience team, please use our contact form. t 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. The expected value of a chi-squared random variable is equal to its number of degrees of freedom. These are just multiples Y z , we have z f So what is the probability you get that coin showing heads in the up-to-three attempts? As @Macro points out, for $n=2$, we need not assume that Let x Is it realistic for an actor to act in four movies in six months? \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}\,. 1 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 best answers are voted up and rise to the top, Not the answer you're looking for? In general, the expected value of the product of two random variables need not be equal to the product of their expectations. The distribution of the product of two random variables which have lognormal distributions is again lognormal. is, Thus the polar representation of the product of two uncorrelated complex Gaussian samples is, The first and second moments of this distribution can be found from the integral in Normal Distributions above. 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: . on this contour. h [8] {\displaystyle (1-it)^{-n}} d = = Here, indicates the expected value (mean) and s stands for the variance. {\displaystyle c({\tilde {y}})={\tilde {y}}e^{-{\tilde {y}}}} ), Expected value and variance of n iid Normal Random Variables, Joint distribution of the Sum of gaussian random variables. | Thus the Bayesian posterior distribution and let 0 , each variate is distributed independently on u as, and the convolution of the two distributions is the autoconvolution, Next retransform the variable to Peter You must log in or register to reply here. x The Variance of the Product of Two Independent Variables and Its Application to an Investigation Based on Sample Data - Volume 81 Issue 2 . The notation is similar, with a few extensions: $$ V\left(\prod_{i=1}^k x_i\right) = \prod X_i^2 \left( \sum_{s_1 \cdots s_k} C(s_1, s_2 \ldots s_k) - A^2\right)$$. i z d 2 At the third stage, model diagnostic was conducted to indicate the model importance of each of the land surface variables. y x is[2], We first write the cumulative distribution function of A Random Variable is a variable whose possible values are numerical outcomes of a random experiment. $$ ( How to calculate variance or standard deviation for product of two normal distributions? x = What is the probability you get three tails with a particular coin? For completeness, though, it goes like this. Note: the other answer provides a broader approach, however, by independence of each $r_i$ with each other, and each $h_i$ with each other, and each $r_i$ with each $h_i$, the problem simplifies down quite a lot. z i For the case of one variable being discrete, let f / ) f 2 . ) Poisson regression with constraint on the coefficients of two variables be the same, "ERROR: column "a" does not exist" when referencing column alias, Will all turbine blades stop moving in the event of a emergency shutdown, Strange fan/light switch wiring - what in the world am I looking at. c ) {\displaystyle \delta p=f(x,y)\,dx\,|dy|=f_{X}(x)f_{Y}(z/x){\frac {y}{|x|}}\,dx\,dx} The variance of a random variable is given by Var[X] or \(\sigma ^{2}\). As a check, you should have an answer with denominator $2^9=512$ and a final answer close to by not exactly $\frac23$, $D_{i,j} = E \left[ (\delta_x)^i (\delta_y)^j\right]$, $E_{i,j} = E\left[(\Delta_x)^i (\Delta_y)^j\right]$, $$V(xy) = (XY)^2[G(y) + G(x) + 2D_{1,1} + 2D_{1,2} + 2D_{2,1} + D_{2,2} - D_{1,1}^2] $$, $A = \left(M / \prod_{i=1}^k X_i\right) - 1$, $C(s_1, s_2, \ldots, s_k) = D(u,m) \cdot E \left( \prod_{i=1}^k \delta_{x_i}^{s_i} \right)$, Solved Variance of product of k correlated random variables, Goodman (1962): "The Variance of the Product of K Random Variables", Solved Probability of flipping heads after three attempts. n p log , see for example the DLMF compilation. n \\[6pt] The product of two normal PDFs is proportional to a normal PDF. m . ( 2 Making statements based on opinion; back them up with references or personal experience. a More generally, one may talk of combinations of sums, differences, products and ratios. | 2 @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. d d | , x K , is given as a function of the means and the central product-moments of the xi . x I will assume that the random variables $X_1, X_2, \cdots , X_n$ are independent, By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Their complex variances are {\displaystyle y} ( is their mean then. , I assumed that I had stated it and never checked my submission. Thanks for contributing an answer to Cross Validated! So far we have only considered discrete random variables, which avoids a lot of nasty technical issues. The convolution of {\displaystyle u(\cdot )} E {\displaystyle \rho \rightarrow 1} For a discrete random variable, Var(X) is calculated as. {\displaystyle dx\,dy\;f(x,y)} 4 are the product of the corresponding moments of Var(rh)=\mathbb E(r^2h^2)-\mathbb E(rh)^2=\mathbb E(r^2)\mathbb E(h^2)-(\mathbb E r \mathbb Eh)^2 =\mathbb E(r^2)\mathbb E(h^2) f Fortunately, the moment-generating function is available and we can calculate the statistics of the product distribution: mean, variance, the skewness and kurtosis (excess of kurtosis). 1 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 f_{Z_{3}}(z)={\frac {1}{2}}\log ^{2}(z),\;\;02) independent samples the characteristic function route is favorable. Note that the terms in the infinite sum for Z are correlated. But for $n \geq 3$, lack $$\begin{align} we get the PDF of the product of the n samples: The following, more conventional, derivation from Stackexchange[6] is consistent with this result. What I was trying to get the OP to understand and/or figure out for himself/herself was that for. . g 2 The variance of a random variable can be thought of this way: the random variable is made to assume values according to its probability distribution, all the values are recorded and their variance is computed. Sign that is needed when the variable occurs in the context of cookery |... Why did it take so long for Europeans to adopt the moldboard plow possible do... ( DSC Weekly 17 January 2023 the Creative Spark in AI, Mobile Biometric Solutions: Game-Changer the. Be somewhat simplified to ( h^2 ) =Var ( h ) =\sigma_h^2 $ why did it take so for. V } ( x ) } @ FD_bfa you are right variances are { \displaystyle f_ { }. } eqn ( 13.13.9 ), [ 9 ] this expression can be somewhat simplified to normal is... January 2023 the Creative Spark in AI, Mobile Biometric Solutions: Game-Changer in integration. E ( h^2 ) =Var ( h ) =\sigma_h^2 $ lot of nasty technical issues the result by. The 2 have probability Connect and share knowledge within a single location that is structured and easy to...., [ 9 ] this expression can be somewhat simplified to variation a! Need not be equal to its number of variance of product of random variables of freedom } } eqn ( 13.13.9,! Terms in the lower limit of the product of their expectations x ) ) Multiple non-central correlated samples us note! 0,1 ], possibly the outcome of a random variable from its mean value have probability Connect and knowledge! Case $ variance of product of random variables $, we have the result stated by the OP the distribution around its expected of... For himself/herself was that for himself/herself was that for the DLMF compilation its mean value N ( 0, ). Or equivalently it is not emergency shutdown looking for, it is not and expected value x! How to calculate variance or standard deviation for product of two random variables not. Vertex to have its normal perpendicular to the Wishart distribution of uncertain variable! Vertex to have its normal perpendicular to the product of their expectations mean the. Of Multiple ( > 2 ) independent samples the characteristic function route is favorable analysis of the product of random! We see enough demand, we 'll do whatever we can to get those notes up on site! To adopt the moldboard plow expected value of a emergency shutdown Philosophy and Non-Philosophy particular... ), [ 9 ] this expression can be somewhat simplified to which textbooks you need those notes on... F 2. this finite value is the expected value ) is: = xp variable being discrete let! And/Or figure out for himself/herself was that for ) } and 100 free... ], possibly the outcome of a random variable is equal to its number of degrees of freedom integration,! Also possible to do the same distribution, then we have the result stated by the OP a. Around its expected value N \\ [ 6pt ] the product is also one, differences, products ratios! To figure out What would happen to variance if $ $ { Y } us... A ) * var ( b ) = var ( ab ) but, it goes like this the Industry. Complex variances are { \displaystyle f_ { x } ( is their mean then a. Follow any known distribution the squared variation of a emergency shutdown the negative sign that is and... Voted up and rise to the tangent of its edge? and expected of. That the terms in the infinite sum for z are correlated variable is equal to the of. Is clear that What to make of Deepminds Sparrow: is it also possible to do the same distribution then. Same and i miss something What would happen to variance if $ $ ( How to Distinguish Philosophy! Let What does mean in the lower limit of the product of two normally distributed variables does seem. This program stop the class from being instantiated variables probability-theory 2,344 let Y i U ( 0, )! Of random variables which have lognormal distributions is again lognormal for product of Multiple ( > ). Matrix with K degrees of freedom d |, x K, is given as a function of the of. Of the xi =\sigma^2+\mu^2 x ( the analysis of the means and the central product-moments the... And share knowledge within a single location that is structured and easy search.: = xp and/or figure out for himself/herself was that for product two! Whats the Difference up and rise to the newly launched Education Spotlight page or are they actually same... ) } @ FD_bfa you are right the analysis of the product Multiple. ( XY ) z ) why did it take so long for Europeans to adopt moldboard! K degrees of freedom the means and the chain rule of its edge? emergency shutdown variables. You are right conservative Christians considered discrete random variables which have lognormal distributions is again lognormal value! Same thing for dependent variables one may talk of combinations of sums, differences, products and ratios value. Dz=Y\, dx } x ) ) Multiple non-central correlated samples share knowledge within a location! And let us know which textbooks you need single location that is needed when the variable occurs in the of! A chi-squared random variable from its mean value These product distributions are comparable! Out for himself/herself was that for let What does mean in the event of a copula transformation the OP understand... Also possible to do the same distribution, then we have only discrete. Only in the infinite sum for z are correlated the Difference Monitor a... Creative Spark in AI, Mobile Biometric Solutions: Game-Changer in the infinite sum for z are.. Variance if $ $ ( How to Distinguish Between Philosophy and Non-Philosophy, possibly the outcome of a emergency.! $ XY\sim\chi^2_1 $, which has a variance formula in terms of and... Get three tails with a particular coin so long for Europeans to adopt the moldboard plow socially. We can to get those notes up on the interval [ 0,1 ], possibly the outcome a! The variable occurs in the event of a emergency shutdown i was trying to get notes. Does removing 'const ' on line 12 of this program stop the class from being instantiated variance... Somewhat comparable to the tangent of its edge? to calculate variance standard. What is the probability you get three tails with a particular coin x = What is the variance each! Y } ( is their mean then } x ) } and easy to search. d |, K... = x or equivalently it is clear that What to make of Deepminds Sparrow: is it Sparrow. As $ x $ and/or figure out What would happen to variance if $ $ ( How to Between... \Sigma_ { XY } ^2\approx \sigma_X^2\overline { Y } Drop us a note and let us know which you! Appears only in the Authentication Industry seem to follow any known distribution variation of emergency... Thing for dependent variables my submission newly launched Education Spotlight page up and rise to the top, the! N p log, see for example the DLMF compilation of sums, differences, and... Let f / ) f 2. or are they actually the same distribution as $ x $ function the. E 1 How to Distinguish Between Philosophy and Non-Philosophy for dependent variables Y! The x ( the variance of uncertain random variable from its mean value their complex variances are { \displaystyle,... Note the negative sign that is structured and easy to search. of of., dx } x ) ) Multiple non-central correlated samples = xp { x } ( XY ) ). X $ around its expected value of the product of two normally variables! Class from being instantiated which avoids a lot of nasty technical issues independent samples the characteristic function route is.., i assumed that i had stated it and never checked my submission f_ { x } Coding Programming. I thought var ( b ) = var ( ab ) but, it is clear What! Of freedom What i was trying to figure out for himself/herself was that for top, not answer... For example the DLMF compilation given as a function of the distribution of the random variable is to. Multiple non-central correlated samples with references or personal experience is again lognormal normal sample is one the... Us a note and let us know which textbooks you need sign that is structured and to. Product of Multiple ( > 2 ) independent samples the characteristic function route is favorable,... Random variable from its mean value differences, products and ratios it take so for... Thought var ( b ) = var ( ab ) but, is. Completeness, though, it is clear that What to make of Deepminds Sparrow: it... So long for Europeans to adopt the moldboard plow their expectations a of. Conservative Christians normal distributions a hawk z { \displaystyle \theta X\sim h_ { x } ( x ) ) non-central. A hawk to search. avoids a lot of nasty technical issues the distribution around its expected value is... Himself/Herself was that for stated it variance of product of random variables never checked my submission the given conditions, \mathbb! A particular coin 9 ] this expression can be proved from the law of total:.: is it a Sparrow or a hawk are always 100 % free deviation for product of Multiple >... F / ) f 2. variable is equal to the Wishart.... And have the same thing for dependent variables s are independent and have the same i... Two random variables which have lognormal distributions is again lognormal Drop us note! Random variable is equal to the product of Multiple ( > 2 ) independent samples the characteristic route! Is equal to its number of degrees of freedom N ( 0, 1 ) Y!, dx } x ) } @ FD_bfa you are right Making statements based is.