Which pairs of random variables are independent?

Which pairs of random variables are independent?

Independence of Random Variables If two random variables, X and Y, are independent, they satisfy the following conditions. P(x|y) = P(x), for all values of X and Y. P(x ∩ y) = P(x) * P(y), for all values of X and Y.

Can two random variables that follow exactly the same distribution be independent of each other?

In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d. or iid or IID.

Can two different random variables have the same distribution?

Any pair of independent random variables X and Y having the same continuous distribution provides a counterexample. In fact, two random variables having the same distribution are not even necessarily defined on the same probability space, hence the question makes no sense in general.

How do you prove independent variables?

You can tell if two random variables are independent by looking at their individual probabilities. If those probabilities don’t change when the events meet, then those variables are independent. Another way of saying this is that if the two variables are correlated, then they are not independent.

What does it mean when two random variables have the same distribution?

p(xi) = 1. We say that two random variables, X and Y , have the same distribution (or are equal in distribution) if they have the same p.m.f. We say that two random variables are equal, and write X = Y , if for all s in S, X(s) = Y (s).

Can you check the independence of two random variables?

Checking the independence of all possible couples of events related to two random variables can be very difficult. This is the reason why the above definition is seldom used to verify whether two random variables are independent. The following criterion is more often used instead.

Which is an example of a dependent random variable?

In Example 2, both the random variables are dependent . Thus the mean of the sum of a student’s critical reading and mathematics scores must be different from just the sum of the expected value of first RV and the second RV. But the answer says the mean is equal to the sum of the mean of the 2 RV, even though they are independent.

How are independent variables similar to independent events?

The concept of independent random variables is very similar to independent events. Remember, two events $A$ and $B$ are independent if we have $P(A,B)=P(A)P(B)$ (remember comma means and, i.e., $P(A,B)=P(A textrm{ and } B)=P(A cap B)$). Similarly, we have the following definition for independent discrete random variables.

What is the definition of mutually independent random variables?

The definition of mutually independent random variables extends the definition of mutually independent events to random variables.