Are XY and X Y independent random variables?

Are XY and X Y independent random variables?

Two jointly continuous random variables X and Y are said to be independent if fX,Y (x,y) = fX(x)fY (y) for all x,y. X and Y are independent iff fX,Y (x,y) = g(x)h(y) for all x,y for some functions g and h. Proof. If X and Y are independent then you need only take g(x) = fX(x) and h(y) = fY (y).

What is var x Y if X and Y are independent?

Independent Random Variables We have used the fact that Var(X+Y) = Var(X) + Var(Y) for independent random variables X and Y. For independent random variables X and Y, we have E(XY) = E(X)E(Y).

Are X and Y independent random variables justify your answer?

Justify your answer. Yes, X and Y are independent random variables. Here is a simple argument to show that, using the fact that X and Y are independent if the joint mass of X and Y equals the product of the mass of X times the mass of Y .

When two random variables X and Y are independent What is the expression for the variance of their sum V x Y )?

For independent random variables X and Y, the variance of their sum or difference is the sum of their variances: Variances are added for both the sum and difference of two independent random variables because the variation in each variable contributes to the variation in each case.

How do you calculate var x and var y?

Var[X+Y] = Var[X] + Var[Y] + 2∙Cov[X,Y] . Note that the covariance of a random variable with itself is just the variance of that random variable. While variance is usually easier to work with when doing computations, it is somewhat difficult to interpret because it is expressed in squared units.

How do you know if a random variable is independent?

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.

Are two random variables independent?

Independence of Random Variables If X and Y are two random variables and the distribution of X is not influenced by the values taken by Y, and vice versa, the two random variables are said to be independent.

How do you remember independent and dependent variables?

Many people have trouble remembering which is the independent variable and which is the dependent variable. An easy way to remember is to insert the names of the two variables you are using in this sentence in they way that makes the most sense.

How do you know if data is independent?

Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true. You can use the equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.

How to prove that X and Y are independent random variables?

X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) i.e. Proof: • If X and Y are independent random variables and Z =g(X), W = h(Y) then Z, W are also independent. ,X Y ( , )= ( ) X Y (F x y F x F y )

Are $X$ and $Y$ independent?

Since $X$ and $Y$ are the result of independent coin tosses, the two random variables $X$ and $Y$ are independent. On the other hand, in other scenarios, it might be more complicated to show whether two random variables are independent.

How many random variables are independent of a coin toss?

Since X and Y are the result of different independent coin tosses, the two random variables X and Y are independent. Also, note that both random variables have the distribution we found in Example 3.3. We can write = 3 16. We can extend the definition of independence to n random variables.

How do you calculate independent discrete random variables?

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 and B) = P ( A ∩ B) ). Similarly, we have the following definition for independent discrete random variables.