1. Linearity of expectation
Difficulty: ★☆☆\(\mathrm{E}(X)=8\) and \(Y=3X-5\). What is \(\mathrm{E}(Y)\)?
Hint
2. Variance after scaling
Difficulty: ★☆☆\(\operatorname{Var}(X)=6\) and \(Y=7-3X\). What is \(\operatorname{Var}(Y)\)?
Hint
3. Standard deviation after scaling
Difficulty: ★☆☆\(X\) has standard deviation \(2.5\). If \(Y=12-4X\), what is the standard deviation of \(Y\)?
Hint
4. Independent linear combination
Difficulty: ★★☆\(X\) and \(Y\) are independent, with expectations \(4\) and \(9\) and variances \(3\) and \(2\), respectively. Let \(W=2X-3Y+1\). Which pair gives the expectation and variance of \(W\)?
Hint
5. Which rule needs independence?
Difficulty: ★★☆\(\mathrm{E}(X)=6\), \(\mathrm{E}(Y)=4\), \(\operatorname{Var}(X)=9\) and \(\operatorname{Var}(Y)=1\). No information about dependence is given. Which statement is necessarily true?
Hint
6. Repeated variable versus independent copies
Difficulty: ★★☆\(X\) has variance \(4\). Let \(A=3X\) and \(B=X_1+X_2+X_3\), where \(X_1,X_2,X_3\) are independent copies of \(X\). Which pair gives \(\operatorname{Var}(A)\) and \(\operatorname{Var}(B)\)?
Hint
7. Distribution of a normal combination
Difficulty: ★★☆\(X\sim N(15,4)\) and \(Y\sim N(9,5)\) are independent. In this notation the second parameter is the variance. What is the distribution of \(2X-Y\)?
Hint
8. Comparing independent normal variables
Difficulty: ★★☆Independent lengths satisfy \(A\sim N(30,9)\) and \(B\sim N(24,7)\), with variances as the second parameters. Find \(P(A>B+2)\), correct to \(4\) decimal places.
Hint
9. Unknown mean and a normal comparison
Difficulty: ★★★\(X\sim N(\mu,4)\) and \(Y\sim N(10,9)\) are independent, with variances as the second parameters. It is known that \(\mathrm{E}(2X+Y)=26\). Find \(P(X-Y>1)\), correct to \(4\) decimal places.
Hint
10. Shared variables in a comparison
Difficulty: ★★★\(X\sim N(4,9)\) and \(Y\sim N(6,4)\) are independent, with variances as the second parameters. Define \(A=2X+Y\) and \(B=X+3Y\). Find \(P(A>B)\), correct to \(4\) decimal places.