# Joint PDF of dependent random variables

How would you solve the below

Let X and Y be random variables with joint probability density function (PDF) given by:

$$f_{X,Y}(x,y) = \begin{cases}cxy &: x \ge 0, y \ge 0, x + y \le 2\\ 0&: \text{otherwise}\end{cases}$$

(a) Find $c$.

(b) Find $\Bbb E [XY ]$.

• Seems like a homework question! tell us what you've figured out so far and where you are stuck. – Ehsan M. Kermani Oct 11 '13 at 22:24
• Hello, welcome to Math.SE. Thank you for your question! It is good practice on this site to add a bit of information on the context your question came up in, and to share your own work on it. It's also fine if you state that you're completely lost -- the information is helpful for answerers to gauge their answer. For more information on asking a good question on this site, see here. – Lord_Farin Oct 11 '13 at 22:51
• Also, to get to know how I nicely typeset your question, and other information about writing maths at this site, see e.g. here, here, here and here. – Lord_Farin Oct 11 '13 at 22:55
• Please read the FAQ and learn the proper etiquette for posting here. – Stefan Smith Oct 12 '13 at 0:58

Draw a picture. Let $T$ be the triangle with corners $(0,0)$, $(2,0)$, $(0,2)$. We want to choose $c$ so that $$\iint_T cxy \,dy\,dx=1.$$ To evaluate the integral, express it as the iterated integral $$\int_{x=0}^2\left(\int_{y=0}^{2-x} cxy\,dy\right)\,dx.$$
For $E(XY)$, we want $$\iint_T (xy)(cxy)\,dy\,dx.$$
• The answer says what to do. Integrate $xyf_{X,Y}(x,y)$ over the plane, which comes down to integrating $(xy)(cxy)$ over the triangle. By the way, I do not get $1$ for $c$. The first integral is $cx(2-x)^2/2)=c(2x-2x^2 +x^3/2)$. Integrating with respect to $x$ from $0$ to $2$ gives $2c/3$, so $c=3/2$. Do check your work, and please note that I could have made an arithmetical error. – André Nicolas Oct 12 '13 at 1:32
The text passage contains a mathematical problem involving random variables and their joint probability density function (PDF). It asks for the determination of a constant and the expected value of the product of the variables. There is a comment section that provides guidance and feedback on the problem.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Joint PDF of dependent random variables

How would you solve the below

Let X and Y be random variables with joint probability density function (PDF) given by:

$$f_{X,Y}(x,y) = \begin{cases}cxy &: x \ge 0, y \ge 0, x + y \le 2\\ 0&: \text{otherwise}\end{cases}$$

(a) Find $c$.

(b) Find $\Bbb E [XY ]$.

• Seems like a homework question! tell us what you've figured out so far and where you are stuck. – Ehsan M. Kermani Oct 11 '13 at 22:24
• Hello, welcome to Math.SE. Thank you for your question! It is good practice on this site to add a bit of information on the context your question came up in, and to share your own work on it. It's also fine if you state that you're completely lost -- the information is helpful for answerers to gauge their answer. For more information on asking a good question on this site, see here. – Lord_Farin Oct 11 '13 at 22:51
• Also, to get to know how I nicely typeset your question, and other information about writing maths at this site, see e.g. here, here, here and here. – Lord_Farin Oct 11 '13 at 22:55
• Please read the FAQ and learn the proper etiquette for posting here. – Stefan Smith Oct 12 '13 at 0:58

Draw a picture. Let $T$ be the triangle with corners $(0,0)$, $(2,0)$, $(0,2)$. We want to choose $c$ so that $$\iint_T cxy \,dy\,dx=1.$$ To evaluate the integral, express it as the iterated integral $$\int_{x=0}^2\left(\int_{y=0}^{2-x} cxy\,dy\right)\,dx.$$
For $E(XY)$, we want $$\iint_T (xy)(cxy)\,dy\,dx.$$
• The answer says what to do. Integrate $xyf_{X,Y}(x,y)$ over the plane, which comes down to integrating $(xy)(cxy)$ over the triangle. By the way, I do not get $1$ for $c$. The first integral is $cx(2-x)^2/2)=c(2x-2x^2 +x^3/2)$. Integrating with respect to $x$ from $0$ to $2$ gives $2c/3$, so $c=3/2$. Do check your work, and please note that I could have made an arithmetical error. – André Nicolas Oct 12 '13 at 1:32
user
The text passage contains a mathematical problem involving random variables and their joint probability density function (PDF). It asks for the determination of a constant and the expected value of the product of the variables. There is a comment section that provides guidance and feedback on the problem.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS