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

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

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

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

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

Find $\text{E}(R)$ and $\text{Var}(R)$.

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

Find $E(2X - Y + 3Z)$ and $\text{Var}(2X - Y + 3Z)$.

Let $X$, $Y$, and $Z$ be independent random variables with standard deviations $\sigma_X = 2$, $\sigma_Y = 1$, $\sigma_Z = 3$ and expectations $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(Z) = 0$.

Find $\text{E}(X + Y + Z)$ and $\text{Var}(X + Y + Z)$.

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

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

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

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

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

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