IB Mathematics Applications & Interpretation 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 $\mathrm{Var}(X)=2$ and $\mathrm{Var}(Y)=3$.

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

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 $\text{Var}(X+2Y)$.