Talk:Distribution of the product of two random variables

The main definition given in this article is not the usual definition, which is that $p$ is a product distribution (over $\Omega _{1}\times \dots \times \Omega _{n}$) if

$p(x_{1},\dots ,x_{n})=p_{1}(x_{1})p_{2}(x_{2})\dots p_{n}(x_{n})$

for all $(x_{1},\dots ,x_{n})$.

Requested move 30 June 2021

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved. (closed by non-admin page mover) Vpab15 (talk) 14:01, 31 July 2021 (UTC)

---

Product distribution → Distribution of the product of two random variables – The "product distribution" with this meaning is a seldom used and idiosyncratic terminology. It is not used because it conflicts with the established Product measure, which defines a "product distribution" by virtue of distributions being measures. --129.16.47.174 (talk) 13:44, 30 June 2021 (UTC)

I concur. The use of "product distribution" to denote the distribution of the multiplication of random variables is very niche; the accepted/standard meaning is as discussed above (product measure). The current choice is, to me, as standard as using "exponential distribution" to denote the distribution of the exponentiation of a random variable. At the very least, there should be a disambiguation linked to Product measure (and the last section about "Use in Theoretical computer science" should either be removed or cleaned up, as it refers to... product distributions in the sense above, of product measures. Overall, the current article is *really* confusing, and I strongly second the requested move.

Clément Canonne (talk) 23:39, 30 June 2021 (UTC)

• Move to Product distribution (statistics). Agree that the title needs to be disambiguated and the current title pointed to an appropriate primary topic, with a hatnote there. I would think that this primary topic is probably not a mathematical subject at all, perhaps Distribution (marketing). But the attempt at natural disambiguation viz Distribution of the product of two random variables is unrecognisable to the general audience. Andrewa (talk) 17:26, 10 July 2021 (UTC)

No. The issue is that a `product distribution` is something else than what is described in this article, so Product distribution (statistics) is not a solution. But one could simply say Distribution of a product (statistics) to have less cognitive load in the lemma. 85.226.193.48 (talk) 20:17, 15 July 2021 (UTC)

I agree. Even within mathematics, what this page is currently describing is *not* what most people think of when they hear `product distribution`. This is basically the same as if the page on geometric distribution were describing a probability distribution over circles, triangles, and squares in the plane as if it were the standard meaning. This is bonkers. Clément Canonne (talk) 04:01, 24 July 2021 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Variance of the product of independent random variables

The formula is wrong for uncorrelated random variables X and Y. A reference is not given. Note that $\operatorname {E} (X^{2}Y^{2})=\operatorname {E} (X^{2})\cdot \operatorname {E} (Y^{2})$ is not generally valid for uncorrelated random variables. — Preceding unsigned comment added by Sigma^2 (talk • contribs) 10:19, 30 July 2020 (UTC) Sigma^2 (talk) 17:40, 30 July 2020 (UTC)

If one applies the FOIL method to the binomial term, the "L" term $\mu _{x}^{2}\mu _{y}^{2}$ is cancelled out by the substraction, is it not? Kylebgorman (talk) 17:47, 26 November 2020 (UTC)

The binomial term is the problem. Sigma^2 (talk) 11:23, 7 May 2021 (UTC)

The variance of $XY$ is

$\operatorname {Var} (XY)=\operatorname {E} [(XY-\operatorname {E} (XY))^{2}]=\operatorname {E} (X^{2}Y^{2})-(\operatorname {E} (XY))^{2}.$

If $X$ and $Y$ are uncorrelated, it follows that $\operatorname {E} (XY)=\operatorname {E} (X)\operatorname {E} (Y)$ and therefore

$\operatorname {Var} (XY)=\operatorname {E} (X^{2}Y^{2})-\mu _{X}^{2}\mu _{Y}^{2}.$

If, additionally (!), $X^{2}$ and $Y^{2}$ are assumed to be uncorrelated, it follows that $\operatorname {E} (X^{2}Y^{2})=\operatorname {E} (X^{2})\operatorname {E} (Y^{2})$ and therefore

$\operatorname {Var} (XY)=\operatorname {E} (X^{2})\operatorname {E} (Y^{2})-\mu _{X}^{2}\mu _{Y}^{2}=(\sigma _{X}^{2}+\mu _{X}^{2})(\sigma _{Y}^{2}+\mu _{Y}^{2})-\mu _{X}^{2}\mu _{Y}^{2}.$

This is the formula wrongly stated in the article for uncorrelated random variables. This formula is true for stochastically independent random variables but, in general, wrong for uncorrelated random variables.

Note that "$X^{2}$ and $Y^{2}$ are uncorrelated" is not implied by "$X$ and $Y$ are uncorrelated". Note further, that "$X$ and $Y$ are stochastically independent" implies both: "$X$ and $Y$ are uncorrelated" and "$X^{2}$ and $Y^{2}$ are uncorrelated".--Sigma^2 (talk) 11:23, 7 May 2021 (UTC)

PS: Compare https://stats.stackexchange.com/questions/15978/variance-of-product-of-dependent-variables for formulas containing $\operatorname {Cov} (X^{2},Y^{2})$ .Sigma^2 (talk) 11:44, 7 May 2021 (UTC)

Error in Diagram

In the "Diagram to illustrate the product distribution of two variables." there is a mistake: dy should be equal to -z/x^2 dx or alternatively -y/x dx (just as stated in the text), but it says "-y/x^2 dx", which is wrong. — Preceding unsigned comment added by 77.190.67.61 (talk) 15:39, 22 April 2021 (UTC)