If $X_1,X_2,X_3$ are three different uniform random variables, calculate $E(X_1 - 2X_2 + X_3)$.

Question

Suppose that a random variable $X_1$ is distributed uniform $[0,1]$, $X_2$ is distributed uniform $[0,2]$ and $X_3$ is distributed uniform $[0,3]$. Assume that they are all independent.
a) Calculate $E(X_1 - 2X_2 + X_3)$.
b) Calculate $E[(X_1 - 2X_2 + X_3)^2]$
c) Calculate $\text{Var}(X_1 - 2X_2 + X_3)$

Any idea or hints on how to figure this out?

Answer 1

For a random variable X uniformly distributed from $[a,b]$, the expected value $E[X]$ is quite simple to calculate.

$E[X]=\dfrac{b+a}{2}$

As mentioned, use the linearity of expectation. $$E[aX+bY]=E[aX]+E[bY]=aE[X]+bE[Y]$$

The variance is given by $Var(X)=\dfrac{1}{12}(b-a)^2$.

If the random variables are all independent, then the covariance is $0$ and $$Var(aX+bY)=Var(aX)+Var(bY)=a^2Var(X)+b^2Var(Y)$$

Answer 2

Any idea or hints?

Recall the basic properties of expectation. Since the random variables follow uniform distributions, the expectations and variances are well known.

a) We have $$E[X_1-2X_2+X_3] = E[X_1]-2E[X_2]+E[X_3].$$

c) Recall the basic properties of variance, and that if $X$ and $Y$ are independent, then $$\text{Var}(X-Y) = \text{Var}(X)+\text{Var}(Y).$$

b) Use a) and c) to solve for b).