A straight forward question on variances

[michonamona]

Homework Statement

suppose you have the following function:

w=a+b(e+z)

a, b and e are constants and z is a random variable distributed by some density function g(z).

What is the variance of w?

i.e. var(w)

Homework Equations

Suppose E(z) = 0 (expectation of z is 0) and var(z)=$$\sigma^{2}$$

The Attempt at a Solution

The solution is var(w)= $$b^{2} \sigma^{2}$$, but I don't understand why. I appreciate your input.

M

 

[Tedjn]

Two facts. First, the variance of a constant plus a random variable is equal to the variance of the random variable. This makes sense if you think of variance as a measure of spread, since adding a constant to every observation doesn't change the spread. Second, the variance of a constant times a random variable is equal to the constant squared times the variance of the random variable. That is, if x is your random variable, var(ax) = a²var(x). Do you understand how the answer follows?

 

[Mark44]

Homework Statement

suppose you have the following function:

w=a+b(e+z)

a, b and e are constants and z is a random variable distributed by some density function g(z).

What is the variance of w?

i.e. var(w)

Homework Equations

Suppose E(z) = 0 (expectation of z is 0) and var(z)=$$\sigma^{2}$$

The Attempt at a Solution

The solution is var(w)= $$b^{2} \sigma^{2}$$, but I don't understand why. I appreciate your input.

M

Let's use caps for random variables to help keep them separate from constants.
From the given information,
E(Z) = 0 and Var(Z) = $\sigma^2$

Also, by definition, Var(Z) = E(Z²) - $\mu^2$.
Since E(Z) = $\mu$ = 0, then Var(Z) = E(Z²).

You're also given that W = a + b(Z + e). Using the properties of expectation, it can be seen that E(W) = a + bE(Z) + be = a + be, since E(Z) = 0.

With all that out of the way, we can tackle Var(W).

Var(W) = E(W²) - (E(W))².

If you replace W with a + b(Z + e) in the first term on the right, and work things through, you get the result you're supposed to get.

 

[statdad]

If X is a random variable and $Y = c + dX$, then

$$Vary(Y) = d^2 Var(X)$$

 

[michonamona]

Thanks guys! I appreciate your help.