Question about Independent Normal Variables

In summary, we are given X and Y as independent normal variables with E(X^4)=3. We are asked to compute Var(X), Var(Y), Var(Z), Cov(X,Y), Cov(Y,Z), Var(X+Y), and Var(X+Z). Using the equations Var(X)=E(X^2)-E(X)^2 and Cov(X,Y)=E[(X-E(X))(Y-E(Y))], we can find that Var(X)=1, Var(Y)=2, and Cov(X,Y)=0. We also know that Var(Z)=E[(X^2)(Y^2)], but it is unclear if E[(X^2)(Y^2)]=E[X^2]E[Y^2].
  • #1
erica1451
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0

Homework Statement


Let X,Y be two independent normal N(0,1) and N(0,2) variables, and Z=XY. Suppose that E(X^4)=3
a) Compute Var(X), Var(Y), Var(Z), Cov(X,Y), Cov(Y,Z), Var(X+Y), Var(X+Z)
b) By computing E[(X^2)(Z^2)], prove that X,Z are not independent.

Homework Equations


Var(X)=E(X^2)-E(X)^2
Cov(X,Y)=E[(X-E(X))(Y-E(Y))]

The Attempt at a Solution


I know that Var(X)=1, Var(Y)=2, Cov(X,Y)=0. The others I'm not sure about. I found that Var(Z)=E[(X^2)(Y^2)] but I'm not sure if E[(X^2)(Y^2)]=E[X^2]E[Y^2]? I'm having similar problems with the other parts.
 
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  • #2
X+Y should be simple. Do you know the variance formula for addition?

I'm not sure if E[(X^2)(Y^2)]=E[X^2]E[Y^2]
Does (X, Y) independent imply that their squares are independent? Can you go from "(X, Y) independent" to "(X^2, Y^2) independent," e.g. by applying FX,Y(x,y) = FX(x)FY(y)?
 
Last edited:

1. What is an independent normal variable?

An independent normal variable is a type of random variable that follows a normal distribution and is not affected by any other variables in a statistical model. This means that the value of the variable does not depend on or influence the values of any other variables in the model.

2. How is an independent normal variable different from a dependent normal variable?

An independent normal variable is not affected by other variables in a statistical model, whereas a dependent normal variable is influenced by other variables in the model. In other words, the value of a dependent normal variable is dependent on the values of other variables in the model.

3. What is the significance of using independent normal variables in statistical analysis?

Using independent normal variables in statistical analysis allows for more accurate and reliable results. It ensures that the values of the variables are not influenced by any other factors, allowing for a better understanding of the relationship between the variables being studied.

4. How are independent normal variables typically represented in statistical models?

Independent normal variables are usually represented by the letter X followed by a subscript or index, such as X1, X2, etc. This helps to differentiate them from other types of variables in the model.

5. Can independent normal variables ever become dependent?

No, independent normal variables will always remain independent and not be influenced by other variables in the model. However, it is important to note that even if variables are considered independent, there may still be other underlying factors that may affect the relationship between them.

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