2 random variables

Question

if you have an equation ex: 2x-4y+8

and are given x is a norm variant with mean  and st dev values given

y has a mean and st dev given.

 

How do you use the mean and st dev of each to get a mean and st dev for the equation

Answer 1

First notation:

E[X] means the expected value of X and equals the mean of X.

E[X^2] means the expected value of X^2 and can be calculated given the mean and standard deviation of X.

Var(X) = E[X^2] - (E[X])^2 and is equal to the square of the standard deviation of X.

It follows that E[X^2] = (standard deviation of X)^2 + (mean of X)^2

 

Similar notation and relationships for Y. 

 

We need to assume that X and Y are independent, otherwise we need the Covariance of XY.

 

Under independence Var (X + Y) = Var(X) + Var(Y).

 

True for scalars:

Var(aX) = a^2*Var(X)

 

True for constants:

Var(a+X) = Var (X)

 

So

Var(2X-4Y+8) = 4*Var(X) + 16 Var(Y). The standard deviation is the square root of the variance.

 

Expected value or mean is more straight-forward.

E[2X-4Y+8] = 2*E[X] - 4*E[Y] + 8