Linear combinations of Normal random variables Quiz

1. $X\sim\mathrm{N}(10,2^2)$ and $Y\sim\mathrm{N}(6,1^2)$ are independent. What is the distribution of $X-Y$?

• A. $\mathrm{N}(4,3)$
• B. $\mathrm{N}(4,5)$
• C. $\mathrm{N}(16,5)$
• D. $\mathrm{N}(4,1)$

2. $X\sim\mathrm{N}(8,2^2)$. What is the distribution of $3X$?

• A. $\mathrm{N}(24,12)$
• B. $\mathrm{N}(24,6)$
• C. $\mathrm{N}(24,36)$
• D. $\mathrm{N}(8,36)$

3. Four independent crates each have mass $\mathrm{N}(25,2^2)$ kg. What is the standard deviation of their total mass?

• A. $4$ kg
• B. $2$ kg
• C. $8$ kg
• D. $16$ kg

4. $X\sim\mathrm{N}(50,4^2)$ and $Y\sim\mathrm{N}(48,3^2)$ are independent. Find $\mathrm{P}(X>Y)$.

• A. $0.345$
• B. $0.775$
• C. $0.500$
• D. $0.655$

5. What condition is needed for $\mathrm{Var}(aX\pm bY)=a^2\mathrm{Var}(X)+b^2\mathrm{Var}(Y)$?

• A. $X$ and $Y$ have the same variance
• B. $X$ and $Y$ are independent
• C. $a$ and $b$ are both positive
• D. $X$ and $Y$ have equal means