Question Type 2: Finding the expected value or variance of a transformed linear combination of multiple variables given some information on their joint summary statistics Exercises

Given $E(X)=5$ and $E(Y)=3$, find $E(4X - 2Y)$.

Let $X$ and $Y$ be independent with $\Var(X)=9$ and $\Var(Y)=4$. Find $\Var(4X - 2Y)$.

If $E(X + 3Y) = 3$ and $E(Y) = 1$, find $E(X)$.

Given independent variables $X$ and $Y$ with $E(X)=2$, $E(Y)=1$, $\Var(X)=16$, and $\Var(Y)=9$, find $E(3X + 5Y)$ and $\Var(3X + 5Y)$.

Let $X$ and $Y$ be independent with $\Var(X)=2$ and $\Var(Y)=3$. Find $\Var(3X - 2Y + 4)$.

Let $X$, $Y$, and $Z$ be independent with $E(X)=2$, $E(Y)=1$, $E(Z)=3$, $\Var(X)=1$, $\Var(Y)=4$, and $\Var(Z)=9$. Find $E(5X - Y + 2Z)$ and $\Var(5X - Y + 2Z)$.

Let $E(X)=1$, $E(Y)=2$, $\Var(X)=4$, $\Var(Y)=9$, and $\Cov(X,Y)=3$. Find $\Var(X + Y)$.

Given $E(X)=0$, $E(Y)=0$, $\Var(X)=9$, $\Var(Y)=16$, and $\Cov(X,Y)=6$, find $\Var(X + 2Y)$.

Suppose $E(X)=2$, $E(Y)=3$, $\Var(X)=1$, $\Var(Y)=2$, and $\Cov(X,Y)=-1$. Find $\Var(2X - Y)$.

Let $X$ and $Y$ have $E(X)=1$, $E(Y)=2$, $\Var(X)=5$, $\Var(Y)=7$, and $\Cov(X,Y)=3$. Define $W = 2X - Y + 4$. Find $E(W)$ and $\Var(W)$.

Suppose $E(X)=3$, $E(Y)=5$, $E(Z)=7$, $\Var(X)=2$, $\Var(Y)=3$, $\Var(Z)=4$, $\Cov(X,Y)=1$, $\Cov(X,Z)=0$, and $\Cov(Y,Z)=2$. Find $\Var(X + Y + Z)$.

Given $E(Y)=2$, $\Var(Y)=9$, and that $X$ is independent of $Y$, you know $\Var(X + 3Y) = 49$. Find $\Var(X)$.