Question Type 1: Finding the expected value or variance of a transformed variable given some information on its summary statistics Exercises

For random variables $X$ and $Y$ with $E(X)=3$ and $E(Y)=4$, determine $E(2X - 4Y + 7)$.

Given a random variable $X$ has $E(X)=4$, calculate $E(5X+3)$.

Given a random variable $X$ has $\mathrm{Var}(X)=2$, calculate $\mathrm{Var}(5X+3)$.

Given $E(X)=2$ and $\mathrm{Var}(X)=3$, find $E(2X-7)$ and $\mathrm{Var}(2X-7)$.

If $X$ and $Y$ are independent with $\mathrm{Var}(X)=6$ and $\mathrm{Var}(Y)=2$, find $\mathrm{Var}(X - 2Y + 5)$.

Let $X$, $Y$, and $Z$ be independent with standard deviations $\sigma_X=2$, $\sigma_Y=1$, $\sigma_Z=3$ and $E(X)=1$, $E(Y)=2$, $E(Z)=0$. Find $E(X+Y+Z)$ and $\mathrm{Var}(X+Y+Z)$.

Random variables $X$ and $Y$ are independent with $E(X)=1$, $E(Y)=2$, $\mathrm{Var}(X)=4$, and $\mathrm{Var}(Y)=9$. Find $E(3X+4Y)$ and $\mathrm{Var}(3X+4Y)$.

Random variables $X$ and $Y$ are independent with standard deviations $\sigma_X=3$ and $\sigma_Y=2$. Given $E(X+3Y)=3$, find $E(3X+9Y)$ and $\mathrm{Var}(3X+9Y)$.

In a portfolio model, $X$ and $Y$ represent returns on two independent assets with $E(X)=0.05$, $E(Y)=0.02$, $\sigma_X=0.1$, and $\sigma_Y=0.08$. An investor's total return is $R=0.6X+0.4Y$. Find $E(R)$ and $\mathrm{Var}(R)$.

Independent random variables $X$, $Y$, and $Z$ satisfy $E(X)=2$, $E(Y)=1$, $E(Z)=3$, $\mathrm{Var}(X)=1$, $\mathrm{Var}(Y)=4$, and $\mathrm{Var}(Z)=9$. Find $E(2X - Y + 3Z)$ and $\mathrm{Var}(2X - Y + 3Z)$.

Random variables $X$ and $Y$ have $\mathrm{Var}(X)=5$, $\mathrm{Var}(Y)=8$, and $\mathrm{Cov}(X,Y)=2$. Compute $\mathrm{Var}(2X - Y)$.

Random variables $X$ and $Y$ satisfy $\sigma_X=3$, $\sigma_Y=4$, and $\rho_{XY}=0.6$. Compute $\mathrm{Var}(4X+5Y)$.