Finding marginal density functions

In summary: The marginal density of X is just the probability density of X. The random variable X may be one component of a much more extensive list (or vector) of properties, but we are just ignoring those other components (if any) when we look at the marginal. In your case, X is the first component of a vector (X,Y), but when we look at the marginal density of X we are ignoring Y. Isn't all this explained in your textbook? If not, I am very surprised.
  • #1
CAF123
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Homework Statement


The joint probability density function of ##X## and ##Y## is given by $$f(x,y) = \frac{1}{8}(y^2 - x^2)e^{-y},\,\,\,\, x \in\,[-y,y]\,\,,y \in\,(0, \infty)$$

Compute the marginal densities of ##X## and ##Y##.

The Attempt at a Solution


I know the defintions are $$ F_X(x) = \int_{- \infty}^{\infty}\,f(x,y)\,dy\, \text{and}\, F_Y(y) = \int_{- \infty}^{\infty}\,f(x,y)\,dx.$$

Am I correct in saying that if the domains of x and y are just numbers then the limits on the integrals are the endpoints of these domains? I.e if in the example above ##x \in\, [2,4], \,\,\text{and}\,\, y\,\in\,[1,3] F_Y(y) = \int_2^4 f(x,y)\,dx,\,\,\text{while}\,\, F_X(x) = \int_1^3 f(x,y)\,dy##, right?

However, in this case, the domain of x depends on y. So x is less than y=x and greater than y = -x. Drawing this, I see it is the portion enclosed by x = |y|, but above the x-axis (since y is greater than 0). Hence, to get the limits when computing ##F_X(x)##, I should say y is from x to infinity. When I check the answers, they have y from |x| to infinity. Can't I replace |x| = x since y is above 0? Or did I miss something else? Many thanks
 
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  • #2
CAF123 said:

Homework Statement


The joint probability density function of ##X## and ##Y## is given by $$f(x,y) = \frac{1}{8}(y^2 - x^2)e^{-y},\,\,\,\, x \in\,[-y,y]\,\,,y \in\,(0, \infty)$$

Compute the marginal densities of ##X## and ##Y##.

The Attempt at a Solution


I know the defintions are $$ F_X(x) = \int_{- \infty}^{\infty}\,f(x,y)\,dy\, \text{and}\, F_Y(y) = \int_{- \infty}^{\infty}\,f(x,y)\,dx.$$

Am I correct in saying that if the domains of x and y are just numbers then the limits on the integrals are the endpoints of these domains? I.e if in the example above ##x \in\, [2,4], \,\,\text{and}\,\, y\,\in\,[1,3] F_Y(y) = \int_2^4 f(x,y)\,dx,\,\,\text{while}\,\, F_X(x) = \int_1^3 f(x,y)\,dy##, right?

However, in this case, the domain of x depends on y. So x is less than y=x and greater than y = -x. Drawing this, I see it is the portion enclosed by x = |y|, but above the x-axis (since y is greater than 0). Hence, to get the limits when computing ##F_X(x)##, I should say y is from x to infinity. When I check the answers, they have y from |x| to infinity. Can't I replace |x| = x since y is above 0? Or did I miss something else? Many thanks

Two points.
(1) You should use the notations ##f_X(x)## instead of ##F_X(x)##, etc., because F looks like a cumulative distribution function. In probability is is common to use small letters for densities or probability mass functions and capital letters for (cumulative) distribution functions.
(2) Always draw a picture when evaluating 2D integrals. If you draw the allowable combinations of x and y, everything will become clear.
 
  • #3
I forgot the portion on the left hand side - it's clear now. I have a conceptual question: what does a marginal density actually mean?
 
  • #4
CAF123 said:
I forgot the portion on the left hand side - it's clear now. I have a conceptual question: what does a marginal density actually mean?

The marginal density of X is just the probability density of X. The random variable X may be one component of a much more extensive list (or vector) of properties, but we are just ignoring those other components (if any) when we look at the marginal. In your case, X is the first component of a vector (X,Y), but when we look at the marginal density of X we are ignoring Y. Isn't all this explained in your textbook? If not, I am very surprised.
 
  • #5
I see. My book did contain that, however it is more geared toward giving examples rather than explaining the theory so that might be why I missed it first time. From this same question, I asks to compute ##E[X]##. I know this is equal to ##\int_{-∞}^{∞} x f_X(x) dx##, so can I write this as ##\int_{-y}^y x f_X(x) dx##. In the book, they take the limits of integration as 0 to ∞. Why is this? I know x is from -y to y, so can't x take on negative values too? (If y = 5, say, then x is from -5 to 5?). It would seem to me that the limits be from -∞ to ∞?
 
  • #6
CAF123 said:
I see. My book did contain that, however it is more geared toward giving examples rather than explaining the theory so that might be why I missed it first time. From this same question, I asks to compute ##E[X]##. I know this is equal to ##\int_{-∞}^{∞} x f_X(x) dx##, so can I write this as ##\int_{-y}^y x f_X(x) dx##. In the book, they take the limits of integration as 0 to ∞. Why is this? I know x is from -y to y, so can't x take on negative values too? (If y = 5, say, then x is from -5 to 5?). It would seem to me that the limits be from -∞ to ∞?

The two integrals
[tex]\int_{-∞}^{∞} x f_X(x) dx [/tex]
and
[tex]\int_{-y}^y x f_X(x) dx [/tex]
are not equal. Why would you think they are the same?
 
  • #7
Ray Vickson said:
The two integrals
[tex]\int_{-∞}^{∞} x f_X(x) dx [/tex]
and
[tex]\int_{-y}^y x f_X(x) dx [/tex]
are not equal. Why would you think they are the same?
The former integral is if x is not restricted to any domain and the latter if it is. No?
 
  • #8
CAF123 said:
The former integral is if x is not restricted to any domain and the latter if it is. No?

No, not in this problem. When you talk about f_X there is no y anywhere; y is gone.
 
  • #9
Ray Vickson said:
No, not in this problem. When you talk about f_X there is no y anywhere; y is gone.
Oh, pardon me, what a mistake to make. The expectation should be a number, not a function. But why are the limits from 0 to ∞?
 
  • #10
CAF123 said:
Oh, pardon me, what a mistake to make. The expectation should be a number, not a function. But why are the limits from 0 to ∞?

I said it before, and I will say it again one more time: draw a diagram! This will show the region in the (x,y) plane where f_{XY}(x,y) > 0.
 
  • #11
I did draw a diagram but the region -y < x < y is the region above y = |x| in the xy plane. That's why I think the limits are from -∞ to ∞ for x. I don't know why x is resticted from 0 to ∞. To get that case, we would need x > |y| I think.
 
  • #12
The function f_X(x) = f(x) is symmetric, so the integral of x*f(x) from -inf to +inf is zero; that is, EX = 0. The integral of x*f from 0 to +inf is positive, so it looks like they are computing (1/2) E|X|.
 
  • #13
Ray Vickson said:
The function f_X(x) = f(x) is symmetric, so the integral of x*f(x) from -inf to +inf is zero; that is, EX = 0. The integral of x*f from 0 to +inf is positive, so it looks like they are computing (1/2) E|X|.

The answer in the back of the book is that EX = 0. Does this mean that the limits are actually -∞ to ∞? When I check the solutions, they compute the integral from 0 to ∞, yet still get 0 (When I work it through, I get 3/4, which as you said is indeed positive).
 
  • #14
CAF123 said:
The answer in the back of the book is that EX = 0. Does this mean that the limits are actually -∞ to ∞? When I check the solutions, they compute the integral from 0 to ∞, yet still get 0 (When I work it through, I get 3/4, which as you said is indeed positive).

You can have EX = 0 without having limits of -∞ and +∞---that is a whole separate issue. The question is: what are the limits in THIS problem? Well, can you have values of x > 100? can you have x < -1000? x > 100,000,000? Is there ANY M > 0 that has P{X > M} = 0 and P(X < -M} = 0? If you think there is, please tell me a suitable value of M. If there is no such value of M, what is that telling you? (I am sure you already know the answer; it just seems to be the case that you lack confidence in your own answers.)
 
  • #15
Ray Vickson said:
You can have EX = 0 without having limits of -∞ and +∞---that is a whole separate issue. The question is: what are the limits in THIS problem? Well, can you have values of x > 100? can you have x < -1000? x > 100,000,000? Is there ANY M > 0 that has P{X > M} = 0 and P(X < -M} = 0? If you think there is, please tell me a suitable value of M. If there is no such value of M, what is that telling you? (I am sure you already know the answer; it just seems to be the case that you lack confidence in your own answers.)
There is no such M. As long as x lies between -y and y. And since y is from 0 to infinity, the value of x can take any value in (-∞, ∞).
 

1. What is a marginal density function?

A marginal density function is a type of probability function that describes the probability of a single variable in a multivariate system. It is derived by integrating the joint probability function over all other variables except the one of interest.

2. How do you find marginal density functions?

To find a marginal density function, you first need to have the joint probability function for all variables in the system. Then, you can integrate the joint probability function over all other variables, leaving only the variable of interest. This will give you the marginal density function for that variable.

3. Why is it important to find marginal density functions?

Finding marginal density functions is important because they allow us to analyze the probability distribution of individual variables in a multivariate system. This can help us understand the relationship between variables and make predictions about future outcomes.

4. What are the assumptions for finding marginal density functions?

The main assumption for finding marginal density functions is that the variables in the system are independent. This means that the probability of one variable does not affect the probability of another variable. Other assumptions may vary depending on the specific method used to find the marginal density function.

5. Can marginal density functions be used for any type of data?

Marginal density functions can be used for any continuous data, as long as the variables are independent. They are commonly used in fields such as statistics, economics, and physics to analyze and model complex systems.

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