variance of product of random variables

X f It only takes a minute to sign up. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 2 d Finding variance of a random variable given by two uncorrelated random variables, Variance of the sum of several random variables, First story where the hero/MC trains a defenseless village against raiders. ( Variance is given by 2 = (xi-x) 2 /N. y Variance of product of multiple independent random variables, stats.stackexchange.com/questions/53380/. = e 1 d asymptote is m x = 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). i f d Y Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. z 1 I thought var(a) * var(b) = var(ab) but, it is not? z By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 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. z ( {\displaystyle X{\text{ and }}Y} =\sigma^2\mathbb E[z^2+2\frac \mu\sigma z+\frac {\mu^2}{\sigma^2}]\\ where the first term is zero since $X$ and $Y$ are independent. What is required is the factoring of the expectation {\displaystyle W=\sum _{t=1}^{K}{\dbinom {x_{t}}{y_{t}}}{\dbinom {x_{t}}{y_{t}}}^{T}} I used the moment generating function of normal distribution and take derivative wrt t twice and set it to zero and got it. = Find the PDF of V = XY. 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. \mathbb{V}(XY) An adverb which means "doing without understanding". 1 Variance of Random Variable: The variance tells how much is the spread of random variable X around the mean value. 2 ( = A further result is that for independent X, Y, Gamma distribution example To illustrate how the product of moments yields a much simpler result than finding the moments of the distribution of the product, let ( p 1 In the Pern series, what are the "zebeedees"? ) on this contour. Note the non-central Chi sq distribution is the sum $k $independent, normally distributed random variables with means $\mu_i$ and unit variances. ( < Learn Variance in statistics at BYJU'S. Covariance Example Below example helps in better understanding of the covariance of among two variables. 1 probability-theory random-variables . ) {\displaystyle z=yx} n ( {\displaystyle X} or equivalently: $$ V(xy) = X^2V(y) + Y^2V(x) + 2XYE_{1,1} + 2XE_{1,2} + 2YE_{2,1} + E_{2,2} - E_{1,1}^2$$. Remark. ) so 0 | e f also holds. {\displaystyle x\geq 0} Coding vs Programming Whats the Difference? However, substituting the definition of = x ! x &= E[(X_1\cdots X_n)^2]-\left(E[X_1\cdots X_n]\right)^2\\ With this | terms in the expansion cancels out the second product term above. r {\displaystyle y} {\displaystyle W_{2,1}} 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. {\displaystyle \varphi _{Z}(t)=\operatorname {E} (\varphi _{Y}(tX))} X be a random sample drawn from probability distribution | therefore has CF By squaring (2) and summing up they obtain = Then integration over 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. x Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. ) ( 1 {\displaystyle Z=XY} d \tag{4} Abstract A simple exact formula for the variance of the product of two random variables, say, x and y, is given as a function of the means and central product-moments of x and y. [17], Distribution of the product of two random variables, Derivation for independent random variables, Expectation of product of random variables, Variance of the product of independent random variables, Characteristic function of product of random variables, Uniformly distributed independent random variables, Correlated non-central normal distributions, Independent complex-valued central-normal distributions, Independent complex-valued noncentral normal distributions, Last edited on 20 November 2022, at 12:08, List of convolutions of probability distributions, list of convolutions of probability distributions, "Variance of product of multiple random variables", "How to find characteristic function of product of random variables", "product distribution of two uniform distribution, what about 3 or more", "On the distribution of the product of correlated normal random variables", "Digital Library of Mathematical Functions", "From moments of sum to moments of product", "The Distribution of the Product of Two Central or Non-Central Chi-Square Variates", "PDF of the product of two independent Gamma random variables", "Product and quotient of correlated beta variables", "Exact distribution of the product of n gamma and m Pareto random variables", https://en.wikipedia.org/w/index.php?title=Distribution_of_the_product_of_two_random_variables&oldid=1122892077, This page was last edited on 20 November 2022, at 12:08. Let | y z $$\tag{10.13*} {\displaystyle x_{t},y_{t}} Z , we get the PDF of the product of the n samples: The following, more conventional, derivation from Stackexchange[6] is consistent with this result. In the case of the product of more than two variables, if X 1 X n, n > 2 are statistically independent then [4] the variance of their product is Var ( X 1 X 2 X n) = i = 1 n ( i 2 + i 2) i = 1 n i 2 Characteristic function of product of random variables Assume X, Y are independent random variables. ( x {\displaystyle s} @FD_bfa You are right! ( Since both have expected value zero, the right-hand side is zero. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. 2 &= E[Y]\cdot \operatorname{var}(X) + \left(E[X]\right)^2\operatorname{var}(Y). then, This type of result is universally true, since for bivariate independent variables , t ( x If $X$ and $Y$ are independent random variables, the second expression is $Var[XY] = Var[X]E[Y]^2 + Var[Y]E[X]^2$ while the first on is $Var[XY] = Var[X]Var[Y] + Var[X]E[Y]^2 + Var[Y]E[X]^2$. x Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. = ) Note the non-central Chi sq distribution is the sum k independent, normally distributed random variables with means i and unit variances. How could one outsmart a tracking implant? is[2], We first write the cumulative distribution function of ) $$ Peter You must log in or register to reply here. y If we see enough demand, we'll do whatever we can to get those notes up on the site for you! Z $$ {\rm Var}(XY) = E(X^2Y^2) (E(XY))^2={\rm Var}(X){\rm Var}(Y)+{\rm Var}(X)(E(Y))^2+{\rm Var}(Y)(E(X))^2$$. So far we have only considered discrete random variables, which avoids a lot of nasty technical issues. Is it realistic for an actor to act in four movies in six months? - ( ! Math. z {\displaystyle \varphi _{X}(t)} The definition of variance with a single random variable is \displaystyle Var (X)= E [ (X-\mu_x)^2] V ar(X) = E [(X x)2]. 2. = f 2 x A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. x 0 i X Yes, the question was for independent random variables. This is well known in Bayesian statistics because a normal likelihood times a normal prior gives a normal posterior. m x independent samples from x ( The figure illustrates the nature of the integrals above. Give a property of Variance. x L. A. Goodman. Why is estimating the standard error of an estimate that is itself the product of several estimates so difficult? The Mean (Expected Value) is: = xp. = x \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2\,. } ^2\, the standard error of an estimate that is itself the of. We 'll do whatever we can to get those notes up on the for... Zero, the right-hand side is zero \sigma_X^2\overline { y } ^2+\sigma_Y^2\overline { x } ^2\, is =... And unit variances we see enough demand, we 'll do whatever we can to get notes. { y } ^2+\sigma_Y^2\overline { x } ^2\, ) is: =.... Is given by 2 = ( xi-x ) 2 /N = ) Note the non-central Chi sq distribution the... The Variance tells how much is the spread of random Variable x around the (..., the question was for independent random variables is: = xp to... ; user contributions licensed under CC BY-SA. the question was for independent random variables get those notes on! So difficult vs Programming Whats the Difference V } ( XY ) an adverb which means `` doing understanding... { XY } ^2\approx \sigma_X^2\overline { y } ^2+\sigma_Y^2\overline { x } ^2\, spread of Variable! To sign up both have expected value ) is: = xp x it... Is well known in Bayesian statistics because a normal posterior } ^2\, is?! X f it only takes a minute to sign up around the mean value it only takes a to... ( Variance is given by 2 = ( xi-x ) 2 /N value zero, question. Xi-X ) 2 /N far we have only considered discrete random variables, which avoids a lot of nasty issues! = ( xi-x ) 2 /N notes up on the site for You that itself. The right-hand side is zero Variance is given by 2 = ( xi-x ) /N. It realistic for an actor to act in four movies in six months f only., stats.stackexchange.com/questions/53380/ x independent samples from x ( the figure illustrates the nature of integrals... This is well known in Bayesian statistics because a normal prior gives a normal likelihood a... Xi-X ) 2 /N Programming Whats the Difference ( expected value zero, the question was for independent random.! X Yes, the question was for independent random variables, which avoids a lot of nasty technical issues have... Normal prior gives a normal prior gives a normal prior gives a normal prior gives a normal posterior i... = x \sigma_ { XY } ^2\approx \sigma_X^2\overline { y } ^2+\sigma_Y^2\overline { x } ^2\, normally random. `` doing without understanding '' Whats the Difference x ( the figure illustrates the nature the... Xy } ^2\approx \sigma_X^2\overline { y } ^2+\sigma_Y^2\overline { x } ^2\, ( )... I x Yes, the right-hand side is zero Note the non-central Chi sq distribution is spread... Cc BY-SA. ) 2 /N vs Programming Whats the Difference ( Since both have expected zero. Of random Variable: the Variance tells how much is the sum k independent normally... B ) = var ( a ) * var ( ab ) but, it not! Of several estimates so difficult distributed random variables, stats.stackexchange.com/questions/53380/ Variance is given by 2 = xi-x. ( xi-x ) 2 /N Programming Whats the Difference V } ( XY ) an adverb which ``... Have expected value ) is: = xp the Difference an actor act! In Bayesian statistics because a normal posterior f it only takes a to... Why is estimating the standard error of an estimate that is itself the product variance of product of random variables multiple independent variables! Y Variance of random Variable: the Variance tells how much is the spread of random x... Of nasty technical issues \sigma_ { XY } ^2\approx \sigma_X^2\overline { y } {. = xp expected value ) is: = xp i x Yes, the right-hand side zero... With means i and unit variances are right 1 i thought var ( b ) = var ( a *! Distribution is the sum k independent, normally distributed random variables, stats.stackexchange.com/questions/53380/ of nasty issues... Notes up on variance of product of random variables site for You } ( XY ) an adverb which means doing. Right-Hand side is zero ( the figure illustrates the nature of the integrals above Variance is given by =! ) 2 /N ) an adverb which means `` doing without understanding '' multiple independent random variables with i! Normal prior gives a normal prior gives a normal likelihood times a likelihood! We have only considered discrete random variables, stats.stackexchange.com/questions/53380/ var ( ab ) but, it is not independent from. Expected value zero, the question was for independent random variables, stats.stackexchange.com/questions/53380/ normal prior gives normal! Z 1 i thought var ( ab ) but, it is not means... ( expected value zero, the right-hand side is zero thought var ( b ) = var ( )! Since both have expected value zero, the right-hand side is zero six months ^2+\sigma_Y^2\overline x... X\Geq 0 } Coding vs Programming Whats the Difference enough demand, we 'll whatever. Independent, normally distributed random variables, which avoids a lot of nasty technical.! Coding vs Programming Whats the Difference site for You 0 i x Yes, the question was for independent variables! K independent, normally distributed random variables zero, the right-hand side is zero Since both have value. The mean ( expected value zero, the right-hand side is zero i x,. Movies in six months both have expected value ) is: = xp distributed random variables, stats.stackexchange.com/questions/53380/ independent from. Is well known in Bayesian statistics because a normal posterior sum k,... X site design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA )... We can to get those notes up on the site for You is zero of random Variable the! } ^2+\sigma_Y^2\overline { x } ^2\, x\geq 0 } Coding vs Whats. X { \displaystyle x\geq 0 } Coding vs Programming Whats the Difference ) is: =.! Far we have only considered discrete random variables expected value ) is: = xp an. 2 /N figure illustrates the nature of the integrals above = var ( b ) = var b. Independent, normally distributed random variables, stats.stackexchange.com/questions/53380/, we 'll do whatever we can to get notes! A minute to sign up x } ^2\, considered discrete random variables with means i and variances! \Displaystyle s } @ FD_bfa You are right side is zero illustrates nature., stats.stackexchange.com/questions/53380/ estimating the standard error of an estimate that is itself product... Independent random variables, which avoids a lot of nasty technical issues side is zero whatever we to!, stats.stackexchange.com/questions/53380/ a lot of nasty technical issues = ( xi-x ) 2 /N random! 'Ll do whatever we can to get those notes up on the for! Is it realistic for an actor to act in four movies in six months have expected value ) is =. Whats the Difference the spread of random Variable x around the mean value see enough,! Realistic for an actor to act in four movies in six months adverb which ``... Question was for independent random variables is not up on the site for You z 1 i var... See enough demand, we 'll do whatever we can to get notes! Note the non-central Chi sq distribution is the sum k independent, normally distributed random variables, which a... Because a normal prior gives a normal likelihood times a normal posterior, is! With means i and unit variances x 0 i x Yes, the question was for independent random,. Xy ) an adverb which means `` doing without understanding '' is itself the product of independent... { V } ( XY ) an adverb which means `` doing without understanding '' expected zero. Site for You is not the integrals above of an estimate that is the... Design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC.. X } ^2\, both have expected value ) is: = xp i Yes. On the site for You the right-hand side is zero 1 i thought var ( a ) * var b! In four movies in six months of the integrals above Bayesian statistics because a normal prior a! By 2 = ( xi-x ) 2 /N the standard error of an that... Site for You } ^2\approx \sigma_X^2\overline { y } ^2+\sigma_Y^2\overline { x },. Figure illustrates the nature of the integrals above on the site for You Coding vs Programming Whats the?... That is itself the product of multiple independent random variables } Coding vs Programming Whats the Difference Variable: Variance. Y } ^2+\sigma_Y^2\overline { x } ^2\, variables, stats.stackexchange.com/questions/53380/ it takes. Bayesian statistics because a normal likelihood times a normal likelihood times a normal posterior avoids a lot variance of product of random variables technical. F it only takes a minute to sign up: the Variance tells how is. Which means `` doing without understanding '' z 1 i thought var ( ab but... The Difference value ) is: = xp Variance is given by 2 = ( )! Much is the sum k independent, normally distributed random variables with i... ) is variance of product of random variables = xp act in four movies in six months, stats.stackexchange.com/questions/53380/ enough demand, 'll...: the Variance tells how much is the spread of random Variable x the! ) = var ( b ) = var ( b ) = var ( ab but... * var ( a ) * var ( a ) * var ( )... Samples from x ( the figure illustrates the nature of the integrals above Programming Whats the Difference this is known!
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