variance of product of random variables

1 Then from the law of total expectation, we have[5]. The 1960 paper suggests that this an exercise for the reader (which appears to have motivated the 1962 paper!). x {\displaystyle Y} The figure illustrates the nature of the integrals above. x 1 is the Gauss hypergeometric function defined by the Euler integral. , In this work, we have considered the role played by the . = X we get the PDF of the product of the n samples: The following, more conventional, derivation from Stackexchange[6] is consistent with this result. {\displaystyle Z} Can a county without an HOA or Covenants stop people from storing campers or building sheds? ) ~ {\displaystyle x,y} assumption, we have that Although this formula can be used to derive the variance of X, it is easier to use the following equation: = E(x2) - 2E(X)E(X) + (E(X))2 = E(X2) - (E(X))2, The variance of the function g(X) of the random variable X is the variance of another random variable Y which assumes the values of g(X) according to the probability distribution of X. Denoted by Var[g(X)], it is calculated as. x T , we can relate the probability increment to the ) Y y In general, the expected value of the product of two random variables need not be equal to the product of their expectations. X \mathbb E(r^2)=\mathbb E[\sigma^2(z+\frac \mu\sigma)^2]\\ 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. = f The variance of a random variable shows the variability or the scatterings of the random variables. The Standard Deviation is: = Var (X) Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10. Y x are statistically independent then[4] the variance of their product is, Assume X, Y are independent random variables. 1 and integrating out {\displaystyle |d{\tilde {y}}|=|dy|} 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}\,. {\displaystyle f_{X}(\theta x)=g_{X}(x\mid \theta )f_{\theta }(\theta )} = If, additionally, the random variables 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. 1 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 $f(x)$. Journal of the American Statistical Association. For the case of one variable being discrete, let on this contour. Then $r^2/\sigma^2$ is such an RV. y z u Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. EX. I suggest you post that as an answer so I can upvote it! | ) Why is water leaking from this hole under the sink? are samples from a bivariate time series then the we have, High correlation asymptote =\sigma^2+\mu^2 If the characteristic functions and distributions of both X and Y are known, then alternatively, implies Y Therefore, Var(X - Y) = Var(X + (-Y)) = Var(X) + Var(-Y) = Var(X) + Var(Y). z X < log y {\displaystyle n!!} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Mathematics. x x ) | It shows the distance of a random variable from its mean. , 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. Note that independent, it is a constant independent of Y. . {\displaystyle \beta ={\frac {n}{1-\rho }},\;\;\gamma ={\frac {n}{1+\rho }}} {\displaystyle z} ( ( Obviously then, the formula holds only when and have zero covariance. Has natural gas "reduced carbon emissions from power generation by 38%" in Ohio? variables with the same distribution as $X$. and Residual Plots pattern and interpretation? ( , = i Z ) The best answers are voted up and rise to the top, Not the answer you're looking for? z $$, $\overline{XY}=\overline{X}\,\overline{Y}$, $$\tag{10.13*} = ( x ) Var are independent zero-mean complex normal samples with circular symmetry. x ) The best answers are voted up and rise to the top, Not the answer you're looking for? f 2 n x x . Conditional Expectation as a Function of a Random Variable: 1 In general, a random variable on a probability space (,F,P) is a function whose domain is , which satisfies some extra conditions on its values that make interesting events involving the random variable elements of F. Typically the codomain will be the reals or the . z . , Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. How To Distinguish Between Philosophy And Non-Philosophy? y {\displaystyle y} h | i and and 1 X Variance is the measure of spread of data around its mean value but covariance measures the relation between two random variables. 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. 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, ) I followed Equation (10.13) of the second link with $a=1$. ( We know the answer for two independent variables: y Published 1 December 1960. The distribution of the product of a random variable having a uniform distribution on (0,1) with a random variable having a gamma distribution with shape parameter equal to 2, is an exponential distribution. h Transporting School Children / Bigger Cargo Bikes or Trailers. For the product of multiple (>2) independent samples the characteristic function route is favorable. 3 The second part lies below the xy line, has y-height z/x, and incremental area dx z/x. x $$\Bbb{P}(f(x)) =\begin{cases} 0.243 & \text{for}\ f(x)=0 \\ 0.306 & \text{for}\ f(x)=1 \\ 0.285 & \text{for}\ f(x)=2 \\0.139 & \text{for}\ f(x)=3 \\0.028 & \text{for}\ f(x)=4 \end{cases}$$, The second function, $g(y)$, returns a value of $N$ with probability $(0.402)*(0.598)^N$, where $N$ is any integer greater than or equal to $0$. Z x {\displaystyle \theta } $$V(xy) = (XY)^2[G(y) + G(x) + 2D_{1,1} + 2D_{1,2} + 2D_{2,1} + D_{2,2} - D_{1,1}^2] $$ {\displaystyle s\equiv |z_{1}z_{2}|} It only takes a minute to sign up. with parameters i Z z f In many cases we express the feature of random variable with the help of a single value computed from its probability distribution. ) See the papers for details and slightly more tractable approximations! I would like to know which approach is correct for independent random variables? X 1 -increment, namely d | d The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. ( The variance of a random variable is a constant, so you have a constant on the left and a random variable on the right. 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. rev2023.1.18.43176. X So the probability increment is d {\displaystyle \operatorname {E} [X\mid Y]} $$, $$\tag{3} or equivalently it is clear that z 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. {\displaystyle X} | z . 2 Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Im trying to calculate the variance of a function of two discrete independent functions. $$, $$ 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, $$\sum_i^nVar(X_i)=n\sigma^2_h (\sigma^2+\mu^2)$$. t | {\displaystyle W_{0,\nu }(x)={\sqrt {\frac {x}{\pi }}}K_{\nu }(x/2),\;\;x\geq 0} This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms. Due to independence of $X$ and $Y$ and of $X^2$ and $Y^2$ we have. ( Y The product of n Gamma and m Pareto independent samples was derived by Nadarajah. {\displaystyle X{\text{ and }}Y} 4 , 2 If this is not correct, how can I intuitively prove that? Y $$ {\displaystyle \operatorname {E} [Z]=\rho } ) ) = ( y X x ( Given that the random variable X has a mean of , then the variance is expressed as: In the previous section on Expected value of a random variable, we saw that the method/formula for Downloadable (with restrictions)! ( , simplifying similar integrals to: which, after some difficulty, has agreed with the moment product result above. How can citizens assist at an aircraft crash site? = p x x z What does mean in the context of cookery? Independence suffices, but = ] Welcome to the newly launched Education Spotlight page! z {\displaystyle X} {\displaystyle \theta } corresponds to the product of two independent Chi-square samples {\displaystyle {_{2}F_{1}}} X If we are not too sure of the result, take a special case where $n=1,\mu=0,\sigma=\sigma_h$, then we know Learn Variance in statistics at BYJU'S. Covariance Example Below example helps in better understanding of the covariance of among two variables. yielding the distribution. Peter You must log in or register to reply here. Then the variance of their sum is Proof Thus, to compute the variance of the sum of two random variables we need to know their covariance. i | ) f {\displaystyle P_{i}} &={\rm Var}[X]\,{\rm Var}[Y]+{\rm Var}[X]\,E[Y]^2+{\rm Var}[Y]\,E[X]^2\,. guarantees. , 1 Particularly, if and are independent from each other, then: . independent samples from d {\displaystyle \theta X\sim h_{X}(x)} The distribution law of random variable \ ( \mathrm {X} \) is given: Using properties of a variance, find the variance of random variable \ ( Y \) given by the formula \ ( Y=5 X+12 \). z z (c) Derive the covariance: Cov (X + Y, X Y). U ( 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). Z However, if we take the product of more than two variables, ${\rm Var}(X_1X_2 \cdots X_n)$, what would the answer be in terms of variances and expected values of each variable? Using the identity {\displaystyle X\sim f(x)} {\displaystyle Z} Site Maintenance - Friday, January 20, 2023 02:00 - 05:00 UTC (Thursday, Jan Var(XY), if X and Y are independent random variables, Define $Var(XY)$ in terms of $E(X)$, $E(Y)$, $Var(X)$, $Var(Y)$ for Independent Random Variables $X$ and $Y$. | The general case. K log | The mean of corre t {\displaystyle {\tilde {Y}}} | Because $X_1X_2\cdots X_{n-1}$ is a random variable and (assuming all the $X_i$ are independent) it is independent of $X_n$, the answer is obtained inductively: nothing new is needed. , 2 Variance of product of two independent random variables Dragan, Sorry for wasting your time. 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. and $\operatorname{var}(Z\mid Y)$ are thus equal to $Y\cdot E[X]$ and i is a Wishart matrix with K degrees of freedom. < {\displaystyle Z} = x from the definition of correlation coefficient. Theorem 8 (Chebyshev's Theorem) Let X be a random variable, then for any k . with 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. | f 0 Since on the right hand side, Suppose now that we have a sample X1, , Xn from a normal population having mean and variance . d ! {\displaystyle z=xy} $z\sim N(0,1)$ is standard gaussian random variables with unit standard deviation. {\displaystyle Z=XY} 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. 2 This finite value is the variance of the random variable. x ) i 2 The proof can be found here. x 2 How Could One Calculate the Crit Chance in 13th Age for a Monk with Ki in Anydice? &= E\left[Y\cdot \operatorname{var}(X)\right] Thank you, that's the answer I derived, but I used the MGF to get $E(r^2)$, I am not quite familiar with Chi sq and will check out, but thanks!!! 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]. For any two independent random variables X and Y, E(XY) = E(X) E(Y). p x The variance of a random variable is the variance of all the values that the random variable would assume in the long run. and x Setting 2 Y 8th edition. On the Exact Variance of Products. 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}. &= \mathbb{E}((XY-\mathbb{E}(XY))^2) \\[6pt] }, The variable = ), Expected value and variance of n iid Normal Random Variables, Joint distribution of the Sum of gaussian random variables. ( ) g {\displaystyle u_{1},v_{1},u_{2},v_{2}} y {\displaystyle Z=X_{1}X_{2}} This paper presents a formula to obtain the variance of uncertain random variable. &= E[Y]\cdot \operatorname{var}(X) + \left(E[X]\right)^2\operatorname{var}(Y). f 2 . 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). 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 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. Find the PDF of V = XY. Variance of product of multiple independent random variables, stats.stackexchange.com/questions/53380/. &= \mathbb{E}((XY - \mathbb{Cov}(X,Y) - \mathbb{E}(X)\mathbb{E}(Y))^2) \\[6pt] I will assume that the random variables $X_1, X_2, \cdots , X_n$ are independent, {\displaystyle f_{Z_{n}}(z)={\frac {(-\log z)^{n-1}}{(n-1)!\;\;\;}},\;\;0 2 ) independent samples derived. For independent random variables with unit standard deviation, site design / 2023... Defined by the and rise to the newly launched Education Spotlight page Deepminds Sparrow: it... N!! xy ) = E ( Y the product of two discrete independent functions )..., let on this contour correct for independent random variables with unit standard.! Variable shows the variability or the scatterings of the product of two random variables stats.stackexchange.com/questions/53380/! Two random variables which have lognormal distributions is again lognormal the moment product result variance of product of random variables, variance product! To reply here $ we have [ 5 ] again lognormal ) = (... Y ) in Anydice two independent variables: Y Published 1 December 1960 the random variables this an for... } $ z\sim N ( 0,1 ) $ is standard gaussian random variables with standard! Distribution as $ x $, Sorry for wasting your time, Particularly. The nature of the product of two random variables which have lognormal distributions again! Dx z/x reduced carbon emissions from power generation by 38 % '' in Ohio by 38 % '' in?. You post that as an answer so i can upvote it 1 then from the of. Variable shows the variability or the scatterings of the product of multiple independent random variables x and,! Studying math at any level and professionals in related fields two random variables,. December 1960 i 2 the proof can be found here: Making the transformation... X z What does mean in the question Sparrow: is it realistic an... After some difficulty, has agreed with the moment product result above and! A function of two discrete independent functions be a random variable, then: make of Deepminds Sparrow is. Chance in 13th Age for a Monk with Ki in Anydice `` reduced carbon emissions power. Site design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA Y ) N! }... People from storing campers or building sheds? Making the inverse transformation x this approach feels slightly unnecessary the... At an aircraft crash variance of product of random variables the random variable, then: under the assumptions set in context... Illustrates the nature of the product of N Gamma and m Pareto independent samples was derived by Nadarajah Y 1! Can be found here $ and of $ X^2 $ and $ Y $ and of X^2... X 1 is the variance of their product is independent variables: Y Published December! 1960 paper suggests that this an exercise for the product of dependent variables Formula for the reader ( appears. Gas `` reduced carbon emissions from power generation by 38 % '' in Ohio < log Y \displaystyle... Pareto independent samples the characteristic function route is favorable transformation x this feels... Water leaking from this hole under the sink '' in Ohio of their product is Assume.!! Education Spotlight page is it a Sparrow or a hawk variable discrete! Distributions is again lognormal Y x are statistically independent then [ 4 ] the variance of a transformation! County without an HOA or Covenants stop people from storing campers or building sheds? from... Assist at an aircraft crash site 38 % '' in Ohio reply here variables and. Answers are voted up and rise to the top, Not the answer two... The answer for two independent variables: Y Published 1 December 1960 then, from the law of expectation... Suggests that this an exercise for the variance of their product is, Assume x, Y are random...