# Joint distribution of functions

• mlarson9000
In summary, the homework statement is that Homework Equations X1 and X2 are independent. Y1=X1+X2, and Y2=X1-X2. Find the density function of Y1.
mlarson9000

## Homework Statement

X1 and X2 are independent~u(0,1)

Y1=X1+X2

Y2=X1-X2

Find the density function of Y1

## Homework Equations

X1=(Y1+Y2)/2
X2=(Y1-Y2)/2

0$$\leq$$Y1$$\leq$$2
-1$$\leq$$Y2$$\leq$$1

0$$\leq$$Y1+Y2$$\leq$$2
0$$\leq$$Y1-Y2$$\leq$$2

-y1$$\leq$$y2$$\leq$$2-y1
-1$$\leq$$y2$$\leq$$y1

## The Attempt at a Solution

I don't understand how to set up the upper and lower bounds for these problems. I have spent the last two days wrestling with this, and I just don't get it. My professor spent almost an hour today trying to explain this to me, and I got nothing out of it. I will send a chocolate chip cookie through the mail to whoever can explain this in a way that I will finally understand. And while you're at it, maybe you can tell me if I should have used whomever in the last sentence, because that's another thing beyond my comprehension.

Problem 1.

I would use the cumulative distribution function. Let F be the cdf for Y1. (Y1 will not be uniform, by the way.)

Then $$F(t)=P(Y_1 \le t)=P(X_1 + X_2 \le t)$$. Now finding $$P(X_1 + X_2 \le t)$$ is a somewhat straightforward problem. Draw $$x_1 x_2$$ coordinate axes. Draw the set of points $$(x_1,x_2)$$ where the pdf for X1 and the pdf for X2 are nonzero (should be a square). Draw the line $$x_1+x_2=t$$ and shade the correct side. Then as a great time-saving trick, instead of integrating, you can find areas using the formula for the area of a triangle (since the joint pdf is constant). There are two cases to consider, depending on what t is (i.e., depending on whether the line $$x_1+x_2=t$$ is below or above the main diagonal of the square).

That's how I would do it. That doesn't look close to your notes at all, though.

Problem 2.

Determine the case of each pronoun by its use in its own clause. The case is not affected by any word outside the clause.

The subject of a clause takes the subjective case, even when the whole clause is the object of a verb or preposition.

In your example, the whole clause is whoever can explain this in a way that I will finally understand.

Whoever is the subject of can explain, so whoever is the correct pronoun, not whomever.

The whole clause whoever can explain this in a way that I will finally understand is the object of the preposition to, but that is irrelevant. The case is not affected by the word to outside the clause.

You had me until you said to draw the line x1+x2=t. What value of do I use t?

mlarson9000 said:
You had me until you said to draw the line x1+x2=t. What value of do I use t?

0.6

Then try it with 1.7

Then try a generic t value.

So 0$$\leq$$Y1$$\leq$$2. The line will have a slope of -1, cutting the square diagonally. It will cut the square in half at t=1. What do I do with this information to get f(y)? F(y)=$$\int$$f(y), so I need to find an equation for the area, and differentiate. This will have two parts. First 0$$\leq$$Y1$$\leq$$1, and then 0$$\leq$$Y1$$\leq$$2. How do I get the second half? Maybe I'll be able to put this together when I'm not sleepy.

## 1. What is a joint distribution of functions?

A joint distribution of functions is a statistical concept that describes the probability distribution of multiple random variables that are related or dependent on each other. It shows how the values of these variables are likely to occur together and provides a comprehensive understanding of their relationship.

## 2. How is a joint distribution of functions different from a single function's distribution?

Unlike a single function's distribution, which describes the probability of a single variable, a joint distribution of functions describes the probability of multiple variables occurring together. It provides a more complete picture of the relationship between the variables and their likelihood of occurring together.

## 3. What are the benefits of using joint distribution of functions in research?

Joint distribution of functions is a useful tool in research as it allows scientists to analyze the relationship between multiple variables and make predictions about their behavior. It can also help identify any dependencies or correlations between variables, which can provide valuable insights for further analysis.

## 4. How is a joint distribution of functions represented graphically?

A joint distribution of functions is typically represented graphically using a scatter plot, where each data point represents a combination of values for the variables. Other graphical representations, such as heat maps or contour plots, can also be used to visualize the relationship between variables.

## 5. Can a joint distribution of functions be used to make predictions?

Yes, a joint distribution of functions can be used to make predictions about the behavior of multiple variables. By analyzing the relationship between the variables, scientists can use the joint distribution to estimate the likelihood of certain outcomes and make informed predictions about future events.

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