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A point $(X, Y)$ is randomly selected from the interior of a square with vertices $(0, 0),(3, 0),(3, 3),(0, 3)$. $Z$ in an area of a rectangle with vertices $(0, 0), (X, 0), (X, Y), (0, Y)$. Determine the distribution of the variable $Z$, expected value of $Z$ and variance of $Z$.

As I understand, we know that there are two random variables. One is $X$, which follows a uniform distribution on the interval $[0,3]$ and the other one is $Y$. It follows a uniform distribution on the same interval, that is $[0,3]$. The two random variables are independent. There is also a rectangle constructed for which the lengths of two adjacent sides are $X$ and $Y$.

We have a following picture - we choose a random point $(X,Y)$ somewhere in a red square:

enter image description here

Therefore we know that $Z = XY$, so:

PDFs of a $X$ and $Y$ with uniform distribution is equal to $\frac{1}{3-0} = \frac{1}{3}$ for $x, y \in [0,3]$ $0$ otherwise.

Therefore PDF of Z is equal to $\frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}$ for $z \in [0,3]$ and $0$ otherwise. PDF is unique for a distribution therefore we have our distribution.

$$E(Z) = E(XY)$$ $$E(Z) = E(X) + E(Y) + Cov(X,Y)$$ We know that variables X and Y are independent, therefore $Cov(X,Y) = 0$ and we have: $$E(Z) = E(X) + E(Y)$$ $E(X) = E(Y) = \frac{0 + 3}{2} = \frac{3}{2}$ and we get that: $$E(Z) = \frac{3}{2} + \frac{3}{2} = 3$$

I don't know how to calculate variance of $Z$.

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  • $\begingroup$ Your distribution of $Z$ is wrong. $\endgroup$
    – balddraz
    Commented May 30, 2023 at 0:16
  • $\begingroup$ @Aruralreader I edited it out. $\endgroup$
    – thefool
    Commented May 30, 2023 at 0:17

2 Answers 2

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I would say

$E[XY] = \int_0^3 \int_0^3 (\frac 13 x)(\frac 13 y) \ dy \ dx = \frac 19 \int_0^3 x\ dx \int_0^3 y\ dy = (\frac {1}{9})(\frac 92)(\frac 92) = \frac {9}{4}$

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Your distribution of $Z = XY$ is wrong. This is a common mistake: the distribution of the product of two random variables is not just the product of their pdfs. You have to formally derive it and it is surprisingly involved for such a simple setup:

Since the support of both $X, Y$ is $(0, 3)$, the support of $Z = XY$ will be $(0, 9)$; so for $z \in (0, 9)$ \begin{align*} F_Z(z) = P[XY \leq z] = P\Big[Y \leq \frac{z}{X}\Big] &= \int\int_{\big\{(x, y) \in [0, 3]^2 \ : \ y \leq z/x \big\}} f_{X, Y}(x, y) \ dydx \\ &= \frac{1}{9}\int_0^{z/3}\int_0^3 \ dydx + \frac{1}{9}\int_{z/3}^3 \int_0^{z/x} \ dydx \\ &= \frac{z}{9} + \frac{z (\log(9) - \log(z))}{9} = \frac{z}{9}\Big[1 - \log\Big(\frac{z}{9}\Big)\Big] \end{align*} and $F_Z(z) = 0, z \leq 0$ and $F_Z(z) = 1, z \geq 9$. Differentiating this cdf we get that $$ f_Z(z) = -\frac{1}{9}\log\Big(\frac{z}{9}\Big) \quad 0 < z < 9 $$ and $0$ elsewhere.

For the expectation and variance, your reasoning, using independence, for the expectation is almost correct. You got $E[Z] = E[X] + E[Y]$ which is a silly mistake because $Z = XY$ was the product not the sum of those variables. So naturally it should instead be $$ E[Z] = \underbrace{ E[XY] = E[X]E[Y] }_{\text{by independence}} = \frac{9}{4} $$ Finally for the variance, you can use the pdf given above or you can use $\text{Var}[Z] = E[Z^2] - E[Z]^2$ along with independence to note that $E[Z^2] = E[X^2Y^2] = E[X^2]E[Y^2]$.

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  • $\begingroup$ Oh, thank you @0XLR ! I checked the article on wikipedia on variance and now I think I get it. You right, in case of expected value I did a stupid mistake. I had to look for a while on your distribution - It's the hardest part to get those integrals set up right. $\endgroup$
    – thefool
    Commented May 30, 2023 at 1:37
  • $\begingroup$ @mathman12 yes setting up the integrals correctly is almost impossible without drawing out the support-region of the variables and the curves; desmos is often a good tool for that $\endgroup$
    – balddraz
    Commented May 30, 2023 at 1:54
  • $\begingroup$ The way that you calculate the PDF of Z - it's not a convolution, right? Convolutions are used in case of sums of random variables. Also, $f_{X,Y}(x,y) = \frac{1}{9}$ because that's the support of z and we know that it sould have uniform distribution, right? $\endgroup$
    – thefool
    Commented May 31, 2023 at 3:27
  • $\begingroup$ @mathman12 $f_{X, Y}(x, y) = f_X(x)f_Y(y) = 1/9$ because of independence; if they weren't independent $(X, Y)$ will not have uniform distribution. Also you may be confusing $Z = XY$, the product of $X$ and $Y$, with their multivariate combination $(X, Y)$. These are different variables and have different supports. $\endgroup$
    – balddraz
    Commented May 31, 2023 at 3:40
  • $\begingroup$ I still have hard time understanding the integrals setup - wouldn't it be enough to integrate the second one? I feel like it should cover the whole support region if $z/3$ can be as close to $3$ as we wish. $\endgroup$
    – thefool
    Commented May 31, 2023 at 3:45

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