Discrete random variables — common mistakes

Always show your working for $E(X)$ and $Var(X)$. While you can use your calculator's statistics mode to verify your final answer, you won't get method marks in an exam without showing the '$\sum x p$' and '$\sum x^2 p$' calculations.

A very common error is forgetting to square the 'a' term when calculating $Var(aX+b)$. Always write $a^2Var(X)$ to remind yourself. Another trap is when 'a' is negative, for example $Var(3-2X)$. Here $a=-2$, so $a^2 = (-2)^2 = 4$. The variance will be $4Var(X)$.

A discrete random variable can only take specific, separate values (like integers 1, 2, 3...). You can count them. A continuous random variable can take any value within a given range (like a person's height, which could be 1.75m, 1.751m, 1.7512m...).

Expectation is the long-run average, not a guaranteed outcome. For a fair die, E(X) = 3.5. This means if you roll it thousands of times, the average of all your scores will be very close to 3.5. It's a theoretical central point of the distribution.

Think of it as 'Mean of the squares minus the square of the mean'. In symbols, that's $E(X^2) - [E(X)]^2$. This verbal cue helps many students remember the order and the squaring.

Absolutely not. A core rule of probability is that for any event A, $0 \le P(A) \le 1$. If you calculate a probability and it's outside this range, you have made a mistake. Similarly, the sum of all probabilities in your distribution must be exactly 1.