Functions of two or more random variables

In summary: I have to compute the prob that pr(Y<y)= pr(x1+x2<y). If we graph y=x1+x2 over the x2 , x1 axes, then we can say that x2=y-x1 and y would be like the shifting value of the line. The defined region is a square and the region of integration would the area defined by this line and the square. I don't understand how to set the boundaries and the integrals. They are confusing me a lot. I would accept an explanation on a scratch paper to see the big picture of it. I would appreciate if
  • #1
TheMathNoob
189
4

Homework Statement


Supposethat X1and X2 are .random variables and that each of them has the uniform distribution on the interval [0, 1]. Find the p.d.f. of Y =X1+X2.

Homework Equations


Find cdf of Y and then the pdf

The Attempt at a Solution


the joint pdf would be f(x1,x2)= 1 0<x1<1 0<x2<1
0 otherwise
so I have to compute the prob that pr(Y<y)= pr(x1+x2<y). If we graph y=x1+x2 over the x2 , x1 axes, then we can say that x2=y-x1 and y would be like the shifting value of the line. The defined region is a square and the region of integration would the area defined by this line and the square. I don't understand how to set the boundaries and the integrals. They are confusing me a lot. I would accept an explanation on a scratch paper to see the big picture of it. I would appreciate if you explain how to approach those kind of exercises that involve two random variables using the definition and the logic of pr(x1+x2<y). The manual solution gives two different pdfs.
 
Physics news on Phys.org
  • #2
Firstly, the problem cannot be done unless we know the dependence between X1 and X2. Almost certainly, the problem wishes you to assume they are independent, but it's very important to get into the habit of noting such things because before long you'll start to get problems where that is not the case, and everything changes. For now, let's assume that they are independent.

Think about the 'isolines' of Y: the subsets of the unit square on which Y is constant. Draw them on a diagram. Now draw the regions of that square where Y<a for any given a in the possible range of Y. With luck, that should give you enough intuition to work out the pdf.
 
  • #3
andrewkirk said:
Firstly, the problem cannot be done unless we know the dependence between X1 and X2. Almost certainly, the problem wishes you to assume they are independent, but it's very important to get into the habit of noting such things because before long you'll start to get problems where that is not the case, and everything changes. For now, let's assume that they are independent.

Think about the 'isolines' of Y: the subsets of the unit square on which Y is constant. Draw them on a diagram. Now draw the regions of that square where Y<a for any given a in the possible range of Y. With luck, that should give you enough intuition to work out the pdf.
I should have stated that the the random variables are independent. Sorry I did not realize that you answered that. I will think about it
 
  • #4
andrewkirk said:
Firstly, the problem cannot be done unless we know the dependence between X1 and X2. Almost certainly, the problem wishes you to assume they are independent, but it's very important to get into the habit of noting such things because before long you'll start to get problems where that is not the case, and everything changes. For now, let's assume that they are independent.

Think about the 'isolines' of Y: the subsets of the unit square on which Y is constant. Draw them on a diagram. Now draw the regions of that square where Y<a for any given a in the possible range of Y. With luck, that should give you enough intuition to work out the pdf.
Can you go over isolines a little bit more?
 
  • #5
TheMathNoob said:
Can you go over isolines a little bit more?
Draw the unit square on a number plane with X1 being the horizontal axis and X2 the vertical. Now draw the line that has all the points where Y=1 (ie X1+X2=1). Then draw another one for Y=1.5 and another for Y=0.5. Those lines are called isolines because for any such line, the value of Y is the same everywhere on the line.
 
  • #6
TheMathNoob said:

Homework Statement


Supposethat X1and X2 are .random variables and that each of them has the uniform distribution on the interval [0, 1]. Find the p.d.f. of Y =X1+X2.

Homework Equations


Find cdf of Y and then the pdf

The Attempt at a Solution


the joint pdf would be f(x1,x2)= 1 0<x1<1 0<x2<1
0 otherwise
so I have to compute the prob that pr(Y<y)= pr(x1+x2<y). If we graph y=x1+x2 over the x2 , x1 axes, then we can say that x2=y-x1 and y would be like the shifting value of the line. The defined region is a square and the region of integration would the area defined by this line and the square. I don't understand how to set the boundaries and the integrals. They are confusing me a lot. I would accept an explanation on a scratch paper to see the big picture of it. I would appreciate if you explain how to approach those kind of exercises that involve two random variables using the definition and the logic of pr(x1+x2<y). The manual solution gives two different pdfs.

Just so you know: cdf's are almost always defined (by modern convention) as ##P(Y \leq y)##, not ##P(Y < y)##. Every book I own adopts this convention, and it is important to pay attention to it (not to just sloppily write "##< y##" when you really mean "##\leq y##"). The reason for being careful is so that more general cases, such as mixed discrete-continuous random variables can be treated properly. Anyway, for ##y \leq 1## the region of relevance is that part of ##\{ x_1 + x_2 \leq y \}## lying in the unit square ##S = \{ (x_1,x_2)\, : \, 0 \leq x_1 \leq 1, 0 \leq x_2 \leq 1 \}##, and the probability (the cdf) will just be the area of that region.

Draw it out for yourself, but here are some hints.
(1) For ##y \leq 1##, what does the region look like? What is its shape? What are its boundaries? What is its area? You know how to draw the line ##x_1 + x_2 = y##; you know you must be on or below that line; and you know you must be inside the unit square. I see no reason why you cannot draw that.
(2) For ##1 < y \leq 2##, what does the region look like, what are its boundaries, and what is its area? Why, for ##y## between 1 and 2 is it easier to look at the complementary region ##\{ x_1 + x_2 > y \}##?
(3) For ##y < 0## what does the region look like? What about for ##y > 1##?
 
  • #7
Ray Vickson said:
Just so you know: cdf's are almost always defined (by modern convention) as ##P(Y \leq y)##, not ##P(Y < y)##. Every book I own adopts this convention, and it is important to pay attention to it (not to just sloppily write "##< y##" when you really mean "##\leq y##"). The reason for being careful is so that more general cases, such as mixed discrete-continuous random variables can be treated properly. Anyway, for ##y \leq 1## the region of relevance is that part of ##\{ x_1 + x_2 \leq y \}## lying in the unit square ##S = \{ (x_1,x_2)\, : \, 0 \leq x_1 \leq 1, 0 \leq x_2 \leq 1 \}##, and the probability (the cdf) will just be the area of that region.

Draw it out for yourself, but here are some hints.
(1) For ##y \leq 1##, what does the region look like? What is its shape? What are its boundaries? What is its area? You know how to draw the line ##x_1 + x_2 = y##; you know you must be on or below that line; and you know you must be inside the unit square. I see no reason why you cannot draw that.
(2) For ##1 < y \leq 2##, what does the region look like, what are its boundaries, and what is its area? Why, for ##y## between 1 and 2 is it easier to look at the complementary region ##\{ x_1 + x_2 > y \}##?
(3) For ##y < 0## what does the region look like? What about for ##y > 1##?
Hi ray, how did you figure out the boundaries of y<=1 and 1<y<=2. I think I have to review calc 3 some more. If you have the stewart book, can you tell me where can I practice this?
 
  • #8
TheMathNoob said:
Hi ray, how did you figure out the boundaries of y<=1 and 1<y<=2. I think I have to review calc 3 some more. If you have the stewart book, can you tell me where can I practice this?

Draw a picture and you will see why!

I don't have the Stewart book. You can find out lots of information by doing the 21st Century version of going to the library---namely, searching on-line. The internet is full of free sites that review all this material, ranging from tutorial pages to entire elementary calculus textbooks.
 
  • #9
Ray Vickson said:
Draw a picture and you will see why!

I don't have the Stewart book. You can find out lots of information by doing the 21st Century version of going to the library---namely, searching on-line. The internet is full of free sites that review all this material, ranging from tutorial pages to entire elementary calculus textbooks.
Hi Ray, now I understand what is going on. If we assume the boundaries to be x1>0 and x2>0 then we will just have a single integral going from 0 to y. In the square region that we have, we just can do this until y=1 otherwise we would get bad areas or areas that don't belong to the region. x2=y-x1 keeps moving until y=2, so we have to find a way to integrate in the interval 1<=y<=2. I came up with the integral 1 - double integral(y-1 to 1 and y-x1 to 1) of 1 dx2 dx1. Is that right?. That's for the cdf
 
Last edited:

Related to Functions of two or more random variables

1. What is the definition of "Functions of two or more random variables"?

A function of two or more random variables is a mathematical expression that combines two or more random variables to produce a new random variable. This new random variable is a function of the original variables and can be used to describe the relationship between them.

2. Why are "Functions of two or more random variables" important in statistics?

"Functions of two or more random variables" are important in statistics because they allow us to model more complex relationships between variables. By combining two or more random variables into a function, we can better understand the behavior of a particular system or phenomenon.

3. How are "Functions of two or more random variables" used in data analysis?

In data analysis, "Functions of two or more random variables" are used to explore the relationship between multiple variables and to make predictions about their behavior. They can also be used to create models and simulations to test different scenarios and make informed decisions.

4. Can "Functions of two or more random variables" be applied to any type of data?

Yes, "Functions of two or more random variables" can be applied to any type of data, including continuous, discrete, and categorical variables. They are a fundamental concept in statistics and can be used in a wide range of applications, such as finance, economics, biology, and engineering.

5. What are some common examples of "Functions of two or more random variables"?

Some common examples of "Functions of two or more random variables" include linear regression, logistic regression, and correlation. These functions are widely used in data analysis to study the relationship between variables and make predictions about future outcomes.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
502
  • Calculus and Beyond Homework Help
Replies
3
Views
495
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
3K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
5
Views
1K
Replies
5
Views
509
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
3K
Back
Top