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

Let $X$ and $Y$ be random variables with $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{Var}(X) = 5$, $\text{Var}(Y) = 7$, and $\text{Cov}(X, Y) = 3$.

Define $W = 2X - Y + 4$. Find $\text{E}(W)$ and $\text{Var}(W)$.

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

Given $E(Y) = 2$ and $\text{Var}(Y) = 9$, let $X$ and $Y$ be independent random variables such that $\text{Var}(X + 3Y) = 130$. Find $\text{Var}(X)$.

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

Let $X$ and $Y$ be independent random variables with $\operatorname{Var}(X)=2$ and $\operatorname{Var}(Y)=3$.

Find $\operatorname{Var}(3X - 2Y + 4)$.

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

Let $X$ and $Y$ be independent random variables with $\text{Var}(X)=9$ and $\text{Var}(Y)=4$. Find $\text{Var}(4X - 2Y)$.

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

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

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

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

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