Joint and conditional distributions

In summary, the conversation discusses the problem of evaluating a distribution with Chi-square random variables X and Y that are independent but not identically distributed. The Bayes' theorem is used to express the distribution as a conditional probability, but difficulties arise in evaluating the distribution due to the random variable Y. The conversation ends with a discussion on finding the "threshold" at which the distribution changes.
  • #1
3
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I'm having a problem evaluating a distribution-

Suppose X and Y are Chi-square random variables, and a is some
constant greater than 0. X and Y are independent, but not identically distributed (they have different DOFs).
I want to find

P(X>a,X-Y>0). So I use Bayes' theorem to write

P(X>a,X-Y>0)
=P(X>a | X-Y > 0)*P(X-Y>0)
=P(X>a| X>Y)*P(X>Y)

Now I have an expression for P(X>a) and P(X>Y), but I am at a
loss as to how to evaluate the conditional distribution P(X>a|
X>Y).

I figured out that if Y was a constant (rather than a random variable), then I could write

P(X>a| X>Y) = { 1 if Y>a
{ P(X>a)/P(X>Y) if Y<a

But this does not help evalaute the distribution because I requires knowledge of the value of random variable Y.

Any help will be much appreciated.
 
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  • #2
Why are you going through the conditional probability formula?

(X>a, X>Y) if and only if (X > max{a,Y}) or (X - max{a,Y} > 0). Just an observation.
 
  • #3
Why are you going through the conditional probability formula?
(X>a, X>Y) if and only if (X > max{a,Y}) or (X - max{a,Y} > 0). Just an observation

Thanks, that is a good observation. So now I know that

P(X>a,X>Y) = P(X>max(a,Y)). I would like to express this as some function of P(X>a) and P(X>Y) . That is, I know that

P(X>a,X>Y) = [ P(X>a) if a>Y
[ P(X>Y) if a<Y

but I only know a, not Y (since Y is an RV). So, in other words, is there a way to determine the 'threshold' at which P(X>a,X>Y) changes from P(X>a) to P(X>Y)?
 
  • #4
BTW, is this homework? If it is, this is not the place to post it.

There is no set threshold because -- as you posted -- Y is a r.v. By implication so is max{a,Y}. I am guessing that the cdf of max{a,Y} would be some linear combination of CDF(Y|Y>a) and the mass point Y=a (representing all the occurances of Y<a). Even after obtaining CDF(max{a,Y}) you still need to figure out the CDF of the related r.v. (X - max{a,Y}).
 

1. What is a joint distribution?

A joint distribution is a probability distribution that describes the likelihood of two or more random variables occurring together. It provides information about the relationship between the variables and how they vary together.

2. How is a joint distribution different from a marginal distribution?

A marginal distribution describes the probability of a single variable occurring, while a joint distribution describes the probability of multiple variables occurring together. Marginal distributions can be obtained by summing or integrating over the other variables in the joint distribution.

3. What is a conditional distribution?

A conditional distribution is a probability distribution that describes the likelihood of one variable occurring given the occurrence of another variable. It provides information about how the variables are related and how the probability of one variable changes with different values of the other variable.

4. How do you calculate a joint distribution?

A joint distribution can be calculated by collecting data on the two or more variables of interest and determining the frequency or probability of each combination of values. This can also be done using statistical software or through mathematical equations.

5. How does understanding joint and conditional distributions help in data analysis?

Joint and conditional distributions provide important information about the relationship between variables and can help identify patterns and trends in the data. This can be useful in making predictions and in understanding the impact of different variables on each other.

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