Discrete random variables

Mastery checklist: after the lecture and again after the tutorial

Use the same checklist twice, matching the two checkpoints in the notes. For each item, explain the condition for using the rule and try an example without opening its working.

1. Explain a random variable as a function assigning a real number to every outcome in the sample space. Distinguish a discrete random variable (finite or countable support) from a continuous one. Continuous variables are contrasted here; their density methods belong to later work.
2. Tabulate a discrete probability law with $f(x)=P(X=x)$, $0\le f(x)\le1$, and $\sum_x f(x)=1$.
3. Find every mode: a value x with the largest probability $P(X=x)$.
4. Find a median m satisfying both $P(X\le m)\ge1/2$ and $P(X\ge m)\ge1/2$. The printed checklist uses $m_1,m_2$ inconsistently in this row; use the same candidate m for both tests. The earlier median discussion explains non-uniqueness and the midpoint convention.
5. Calculate the mean $\mu=E(X)=\sum_x xf(x)$.
6. Calculate variance using $\sigma^2=\operatorname{Var}(X)=E[(X-\mu)^2]=\sum_x(x-\mu)^2f(x)$ or $E(X^2)-[E(X)]^2$, then take the nonnegative square root for standard deviation $\sigma$.
7. For $Y=g(X)$, use $E(Y)=E[g(X)]=\sum_xg(x)f(x)$, combining probabilities when constructing Y’s distribution if different x values give the same Y.
8. Transform moments with $E(aX\pm b)=aE(X)\pm b$ and $\operatorname{Var}(aX\pm b)=a^2\operatorname{Var}(X)$.
9. For n independent observations of the same distribution, calculate $E(X_1+\cdots+X_n)=nE(X)$ and $\operatorname{Var}(X_1+\cdots+X_n)=n\operatorname{Var}(X)$. Additivity of expectation itself does not require independence.
10. For two independent variables X,Y with finite moments, use $E(X\pm Y)=E(X)\pm E(Y)$ and $\operatorname{Var}(X\pm Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$. A minus sign does not subtract independent variances.

Choose practice by the skill you need

The notes’ complete question-type map is retained below with links to the actual questions. The part labels describe where to focus within each question. “First part” and “second part” retain the source’s unnumbered groupings; a dash means the source cell is blank.

The printed map groups some questions broadly. For example, Example 3(ii) is a tail probability using a geometric progression, and Review 7(iii) needs the nonlinear area transformation $A=4\pi S^2$. Follow the actual asks and conditions in the linked question, rather than assuming every entry in a row uses a single formula.

Complete discussion and review sets

The questions and short answer keys at the back of both lecture copies match the existing worked records. Use these links to try each question and then reveal its working. The original question pages and short keys are retained in the source record.

• Discussion: Question 1, Question 2, Question 3, Question 4, Question 5, Question 6, Question 7, Question 8.
• Review: Question 1, Question 2, Question 3, Question 4, Question 5, Question 6, Question 7, Question 8, Question 9, Question 10, Question 11, Question 12.

Discussion 4’s short key labels its final three values “(iv)” although the question ends at (iii); the linked working correctly associates them with the mean, expected squared deviation and variance.

Review 11 calls its dice “tetrahedral (8-sided)”; the eight stated face counts determine the model.

Review 12’s three beam panels are reconstructed beside its question, including both alternating stacks and the initial unbalanced beam. These source qualifications remain beside the worked records.