Linear combinations of random variables — common mistakes

Always be careful with variance vs standard deviation. The formulas use variance. If the question gives you standard deviation, square it first. When you standardise a Normal distribution, you divide by the standard deviation, so you'll need to take the square root of your calculated variance. Also, clearly define your new variable (e.g., 'Let W = 2X + Y') and state its distribution, including its mean and variance, before calculating any probabilities.

Variance measures uncertainty or spread. If you have two uncertain quantities, X and Y, their difference, X - Y, is also uncertain. The total uncertainty comes from the variability in both X and Y. Imagine two people trying to guess a length; the difference between their guesses is likely to be more variable than a single person's guess, not less. Mathematically, $\text{Var}(X - Y) = \text{Var}(X + (-1)Y) = \text{Var}(X) + (-1)^2\text{Var}(Y) = \text{Var}(X) + \text{Var}(Y)$.

No. The formulas $E(aX + bY) = aE(X) + bE(Y)$ and (for independent variables) $\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$ work for any type of random variable, discrete or continuous. However, the property that the resulting distribution is also Normal only applies if the original variables X and Y were Normal.

This is a crucial distinction. '2X' means you take one observation of X and double its value. The variance is $\text{Var}(2X) = 2^2\text{Var}(X) = 4\text{Var}(X)$. 'X₁ + X₂' means you take two independent observations from the same distribution and add their values. The variance is $\text{Var}(X_1 + X_2) = \text{Var}(X_1) + \text{Var}(X_2) = \text{Var}(X) + \text{Var}(X) = 2\text{Var}(X)$. The variance is smaller when you combine two independent observations.