Bivariate normal distribution- converse question

[bobby2k]

bivariate normal distribution-"converse question"

Hello, I have a theoretical question on how to use the bivariate normal distribution. First I will define what I need, then I will ask my question.

pics from: http://mathworld.wolfram.com/BivariateNormalDistribution.html

We define the bivariate normal distribution, (1):

From this we get the marginal distributions:

No comes my question:

Let's say that we have 2 random variables x1 and x2, and we know that each marginal distribution satisfies (2) and(3), that is, we know they are normal, and we know their mean, and variance. Suppose we also know their correlation-coefficient p. How can we now say that equation (1) is the joint probability density function. I mean, we defined it one way, and got the marginals, what is the justification that if we have 2 marginals and their p, we can go back? I mean, it is not allways true that the converse is true, why can we assume the converse here?

They used this technique in my book when proving that [itex]\bar{X}[/itex] and [itex]S^{2}[/itex] are independent.

 

[Stephen Tashi]

http://physicsforums.bernhardtmediall.netdna-cdn.com/images/physicsfor[/b]

How can we now say that equation (1) is the joint probability density function.

A Wikipedia article claims we can't say that in the case [itex] \rho = 0 [/itex]. http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent.

However, it is possible for two random variables X and Y to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent;

Perhaps you should give the exact statement of what your book proves.

 

[bobby2k]

A Wikipedia article claims we can't say that in the case [itex] \rho = 0 [/itex]. http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent.



Perhaps you should give the exact statement of what your book proves.

Thanks for your time, I am going to scan it up, be back later.

 

[bobby2k]

Here I have scanned the proof.
http://i.imgur.com/naRsk9s.jpg

The proof starts inside the green line, and the quote I am interested in starts inside the red line. I have however added some information that is before this, so you can see where it all comes from.

 

[blue_raver22]

Hello, thank you for your question. Let me start by clarifying that the bivariate normal distribution is a probability distribution that describes the relationship between two continuous random variables. It is defined by the joint probability density function (PDF) given in equation (1) in your question. This means that when we have two random variables x1 and x2 that follow this distribution, the probability of them taking on certain values can be calculated using this equation.

Now, to address your question about the converse. The converse of a statement is when the order of the original statement is reversed. In this case, the original statement is that if we have two random variables x1 and x2 that follow the bivariate normal distribution, we can use equation (1) to calculate the joint probability density function. The converse of this statement would be that if we have two random variables x1 and x2 that have certain marginals (i.e. follow a normal distribution with known mean and variance) and a known correlation coefficient, we can assume that they follow the bivariate normal distribution.

The justification for this assumption lies in the properties of the bivariate normal distribution. One of the main properties is that the joint distribution of two variables that follow the bivariate normal distribution is completely determined by their marginals and their correlation coefficient. This means that if we know the marginals and the correlation coefficient, we can uniquely determine the joint distribution, which is given by equation (1). Therefore, it is valid to assume that if we have two random variables with known marginals and correlation coefficient, they follow the bivariate normal distribution.

In terms of using this technique to prove independence, it is important to note that the bivariate normal distribution is a special case where the correlation coefficient is 0, meaning that the two variables are independent. This is why it can be used to prove independence in certain cases.

I hope this helps to clarify the concept of the bivariate normal distribution and its converse. Let me know if you have any further questions.