How can I prove X = (X1, X2, X3) is independent?

In summary, the conversation discusses the problem of proving the independence of X = (X1, X2, X3) where X1, X2, and X3 are derived from a set of random variables (Y1, Y2, Y3, Y4) with a certain distribution. The individual values of X1, X2, and X3 are calculated using a formula that involves the random variables and a Gaussian distribution. The speaker considers showing the independence of X1 and X2 and X2 and X3, but it is pointed out that this does not necessarily guarantee the independence of X1 and X3.
  • #1
kkjs
1
0
For (Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5)
let Xk = [∑(from i=1 to k) Yi] / [∑(from i=1 to k+1) Yi] where k = 1,2,3

How can I prove X = (X1, X2, X3) is independent?
What I did was...

(Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5) = (Z1, Z2, Z3, Z4) / (Z1+Z2+Z3+Z4+Z5) where Z ~ N(0,1), Z IID G(1/2)

Now, we have
X1 = Z1 / (Z1+Z2)
X2 = (Z1+Z2) / (Z1+Z2+Z3)
X3 = (Z1+Z2+Z3) / (Z1+Z2+Z3+Z4)

I think if i can somehow show X1 and X2 are independent and X2 and X3 are independent then X1 and X3 are independent as well but how? this is a part I don't get T-T
 
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  • #2


I can't understand your notation, hopefully someone will jump in, but in general, if you're talking about linear independence of vectors, it's not a transitive relation.
 
  • #3


kkjs said:
I think if i can somehow show X1 and X2 are independent and X2 and X3 are independent then X1 and X3 are independent as well but how? this is a part I don't get T-T
That won't work in general. What if, for example, X3 turned out to be a multiple of X1? If X1 and X2 are independent, then X2 and X3 would also be independent, but X1 and X3 are definitely not independent.

Perhaps you can make it work because of the way the X's are constructed in your specific problem, but I'd guess it's not the right strategy.
 

Related to How can I prove X = (X1, X2, X3) is independent?

What is independence in terms of variables?

Independence between variables means that the values of one variable do not affect or influence the values of the other variable. In other words, the occurrence or non-occurrence of one variable does not have any impact on the occurrence or non-occurrence of the other variable.

What does it mean for three variables to be independent?

When three variables, X1, X2, and X3, are independent, it means that the values of X1 do not have any effect on the values of X2 or X3, and the values of X2 do not have any effect on the values of X1 or X3, and the values of X3 do not have any effect on the values of X1 or X2. In other words, the three variables are completely unrelated to each other.

How do I prove that X = (X1, X2, X3) is independent?

In order to prove that X = (X1, X2, X3) is independent, you need to show that the joint probability distribution of X1, X2, and X3 is equal to the product of their individual probability distributions. This means that the probability of X1 and X2 and X3 occurring together is equal to the probability of X1 occurring multiplied by the probability of X2 occurring multiplied by the probability of X3 occurring.

What statistical tests can be used to prove independence?

There are various statistical tests that can be used to prove independence. Some of the commonly used tests include the chi-square test, the Pearson correlation coefficient, and the Spearman correlation coefficient. These tests help to determine the strength of the relationship between variables and can indicate whether the variables are independent or not.

Why is it important to prove independence between variables?

Proving independence between variables is important because it allows us to make reliable predictions and draw accurate conclusions from our data. When variables are independent, we can assume that changes in one variable are not influenced by changes in the other variable, which helps us to better understand the relationships and patterns in our data.

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