Marginal PDF for Uniformly Distributed X and Y?

In summary, Jamie is trying to find the marginal pdfs for X and Y in a joint pdf, but is having trouble calculating the integrals. He suspects that something might be wrong with his bounds but is not sure.
  • #1
JamieL
4
0
Hi all,
I'm looking at the joint pdf F(x,y) = (8+xy^3)/64) for -1<x<1 and -2<y<2
(A plot of it is here: https://www.wolframalpha.com/input/?i=(8+xy^3)/64+x+from+-1+to+1,+y+from+-2+to+2 ...sorry about the ugly url) and trying to find the marginal PDFs for X and Y.

I know I want to integrate the joint function with respect to Y and X in order to to get the marginal pdfs for X and Y, respectively. However, I'm running into trouble when I try to set the bounds for these integrals!
As far as I can tell, X and Y don't seem to depend on each other in this sense; i.e. for marginal(X) i would have Integral([JointPDF]dy), from -2 to 2, which comes out to 1/2.
(Similarly, integrating with respect to x from -1 to 1 yields 1/4).
When I integrate these from their respective bounds (x from -1 to 1, y from -2 to 2) both come out to 1, as a proper pdf should. However the fact that both are independent of x and y values makes me think something might be wrong...does anyone have any suggestions as to what I might be doing wrong?
Thanks so much!
Jamie

Edit: My apologies! Posted this to the wrong place! I can't figure out how to delete it though :x
 
Last edited:
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  • #2
JamieL said:
Hi all,

As far as I can tell, X and Y don't seem to depend on each other in this sense; i.e. for marginal(X) i would have Integral([JointPDF]dy), from -2 to 2, which comes out to 1/2.

How did you get the integral with respect to y to come out to be a constant? It should be a function of x.
 
  • #3
Maybe I'm mistaking, but as far as I can tell the indefinite integral comes out to:

(8y+(xy^4)/4)/64 + c. If you evaluate this from -2 to 2, the x terms cancel because the y is an even function, i.e. from -2 to 2 we get [(1/4)+16x]-[-(1/4)+16x], so (1/4)+(1/4)+16x-16x = 1/2.
This is what made me think that perhaps my bounds are incorrectly calculated...
Am I doing something wrong?
 
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  • #4
JamieL said:
..
Am I doing something wrong?

Not in the integration. (I'm the one who was confused about that.) But the fact that the conditional distributions are constant (and thus "independent" of the values of both variables) doesn't show that the x and y are independent random variables. If x and y were independent random variables then for each pair of sets [itex] A,B [/itex] that define events [itex] Pr(x \in A) = Pr(x \in A | y \in B) [/itex]

For example, consider the events [itex] A = \{x: x \in [0, 1]\},\ B = \{y: y \in [0,1]\} [/itex]
 
  • #5
Still, what would the bounds be? A marginal pdf should not be a constant
 
  • #6
jakegoodman said:
A marginal pdf should not be a constant

I don't know any theorem that asserts that. Do you?
 
  • #7
I think a marginal pdf would be constant if X and Y are uniformly distributed, but I'm not sure how to tell if they are in this case. Any input?
 

What does "Marginal PDF from Joint PDF" mean?

The Marginal PDF from Joint PDF refers to a statistical concept where the probability density function (PDF) of a single random variable is calculated from the joint probability distribution of multiple random variables. Essentially, it allows us to analyze the behavior of one variable without taking into account the other variables.

Why is the Marginal PDF from Joint PDF important?

The Marginal PDF from Joint PDF is important because it allows us to understand the relationship between multiple variables by breaking it down into individual components. It also allows us to make predictions and inferences about a specific variable, without having to consider the others.

How is the Marginal PDF from Joint PDF calculated?

The Marginal PDF from Joint PDF is calculated by integrating the joint probability distribution over all possible values of the other variables, leaving only the desired variable as the output. This can be done using techniques such as double integrals for continuous distributions or summations for discrete distributions.

What is the difference between Marginal PDF and Joint PDF?

The main difference between Marginal PDF and Joint PDF is that Marginal PDF focuses on the probability distribution of a single variable, while Joint PDF considers the probability distribution of multiple variables simultaneously. Marginal PDF is derived from the Joint PDF by integrating out the other variables.

In what situations is the Marginal PDF from Joint PDF used?

The Marginal PDF from Joint PDF is commonly used in statistics, probability, and data analysis to understand the behavior of individual variables in relation to others. It is often used in multivariate analysis, where the relationship between multiple variables needs to be examined and understood.

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