Random variables that are triple-wise independent but quadruple-wise dependent

In summary, the problem is to find four random variables taking values in {-1, 1} such that any three are independent but all four are not. By considering products of independent random variables, we can construct four events of interest that are pairwise independent but not all three are independent. This can be done by using 8 elementary events with probabilities of 1/8 and constructing a table where the four events of interest are independent in pairs and dependent when all four are considered.
  • #1
CantorSet
44
0
Hi everyone, here's a probability problem that seems really counter-intuitive to me:

Find four random variables taking values in {-1, 1} such that any three are independent but all four are not. Hint: consider products of independent random variables.

My thoughts:
From a set perspective, there are no four sets such that any three are disjoint but all four overlap somewhere.

But according to the hint, let [tex]X_{i}[/tex] be a random variable taking values [tex]1[/tex] with probability [tex]p_{i}[/tex] and [tex]-1[/tex] with probability [tex]1-p_{i}[/tex] for all [tex]i \in N[/tex]. Then, I guess our four random variables will be products of these [tex]X_{i}'s[/tex] but how should one construct them?
 
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  • #2
Here's the solution for pairwise independent (3 events) but not all three (you can generalize to 4).

Four elementary events (j,k,l,m), each probability 1/4.
A=(j,k)
B=(j,l)
C=(j,m)
P(A)=P(B)=P(C)=1/2
P(A*B)=P(A*C)=P(B*C)=1/4=P(A)P(B)=P(A)P(C)=P(B)P(C) (independent)
P(A*B*C)=1/4, P(A)P(B)P(C)=1/8 (not independent)
 
  • #3
For your problem work with 8 elementary events (G,H,I,J,K,L,M,N) each with Prob.=1/8 and 4 events of interest(A,B,C,D). Use the following table:
__G-H-I-J-K-L-M-N
A-1-1-1-1-0-0-0-0
B-1-1-0-0-1-1-0-0
C-1-0-1-0-1-0-1-0
D-1-0-0-1-0-1-1-0

1 means in the set, 0 means not in.

If you calculate the probabilities, you will see they are independent in 2's and 3's, but not all 4.
 

1. What is the definition of triple-wise independence?

Triple-wise independence refers to a situation where three random variables are independent of each other. This means that the occurrence of one variable does not affect the occurrence of the other two variables.

2. How is quadruple-wise dependence different from triple-wise independence?

Quadruple-wise dependence means that four random variables are dependent on each other. This is different from triple-wise independence, where only three variables are independent of each other.

3. Can you give an example of random variables that are triple-wise independent but quadruple-wise dependent?

Yes, one example is the height, weight, and shoe size of a group of people. These three variables may be independent of each other, but they are all dependent on the variable of age, making them quadruple-wise dependent.

4. How is the independence of random variables determined?

The independence of random variables is determined by calculating the joint probability distribution of the variables. If the joint probability is equal to the product of the individual probabilities, then the variables are considered independent.

5. Why is it important to understand the concept of triple-wise independence but quadruple-wise dependence?

Understanding this concept is important in statistical modeling and analysis. It allows us to accurately assess the relationships between multiple variables and make more informed decisions based on the data. It also helps in identifying potential confounding factors that may affect the results of a study.

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