Question about Independent Random Variables and iid

In summary, the conversation discusses independent random variables and their properties. The sample space for two flips is defined and the concept of independent identically distributed (iid) random variables is introduced. The discussion then focuses on calculating the probability of events involving these variables and clarifying any confusion. The sample space for n independent random variables is also mentioned.
  • #1
chingkui
181
2
I have a question about independent random variable:
Let say we flip a fair coin, the set of outcome is S={H,T}, P(H)=1/2, P(T)=1/2. Define random variable X:S->R by X(H)=1, X(T)=-1.
From what I read in books, I can define X1 and X2 as independent identically distributed (iid) random variables with the same distribution as X. Then, that would mean P(X1=1,X2=1)=P(X1=1)P(X2=1).
It can easily be seen that the event {X1=1,X2=1}={w in S:X1(w)=1,X2(w)=1}={w in S:w=H}={w in S:X1(w)=1}={X1=1}={w in S:X2(w)=1}={X2=1}, and since P({w:w=H})=P(H)=1/2, we have P(X1=1,X2=1)=1/2 and P(X1=1)P(X2=1)=1/4.
So, I am not sure exactly what I mistake I made. Please help me clear up my confusion. Thanks.
 
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  • #2
The sample space (S) for two flips has four points (H,H), (H,T), (T,H), (T,T), each of which has probability 1/4. In abstract terms, the sample space for n independent random variables is the direct product of the sample spaces of each of the variables.
 
  • #3


Your mistake is in assuming that X1 and X2 are independent random variables. In order for two random variables to be independent, their joint probability distribution must factorize into the product of their individual probability distributions. In this case, P(X1=1,X2=1) does not equal P(X1=1)P(X2=1), as you have correctly calculated. This means that X1 and X2 are not independent, and therefore not iid.

To understand why this is the case, consider the definition of independence: two random variables X and Y are independent if and only if for any events A and B, P(X in A, Y in B) = P(X in A)P(Y in B). In this case, A and B are the events {X=1} and {Y=1}, respectively. However, P(X=1, Y=1) = P(X=1) = 1/2, while P(X=1)P(Y=1) = (1/2)(1/2) = 1/4. Therefore, X and Y are not independent.

To have independent and iid random variables, X1 and X2 would need to have a joint probability distribution that factorizes into the product of their individual probability distributions, which is not the case in this scenario. Therefore, X1 and X2 are not independent and not iid.
 

1. What is the difference between independent random variables and iid (independent and identically distributed) random variables?

Independent random variables are two or more variables that have no influence on each other and have their own probability distributions. On the other hand, iid random variables are a special case of independent random variables where each variable has the same probability distribution. This means that not only are they independent, but they also have the same chance of occurrence.

2. How can we determine if two random variables are independent?

Two random variables are considered independent if the outcome of one variable does not affect the outcome of the other. This can be determined by calculating the joint probability of the two variables and comparing it to the product of their individual probabilities. If they are equal, the variables are independent.

3. Why is it important to know if two random variables are independent or not?

Knowing if two random variables are independent or not is important because it allows us to make certain assumptions and simplifications in statistical analyses. For example, if we know that two variables are independent, we can use simpler models and calculations to analyze their relationship and make predictions.

4. Can two random variables be independent but not identically distributed?

Yes, two random variables can be independent but not identically distributed. This means that although they do not influence each other, they have different probability distributions. This is a more general case of independent random variables and does not have the same simplifications and assumptions as iid random variables.

5. How are independent random variables and iid random variables used in statistical analyses?

Independent random variables and iid random variables are often used in various statistical analyses, such as hypothesis testing, regression analysis, and probability calculations. They allow for simpler and more efficient calculations and assumptions, making it easier to analyze data and make predictions.

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