Statistical Officer / Government Statistical Service (GSS) Assessment Probability and Statistical Inference Syllabus

Probability and Statistical Inference syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Probability and Statistical Inference in Statistical Officer / Government Statistical Service (GSS) Assessment, not a summary of it.

1. 1

   Probability Foundations

   3 topics

   • Basic probability rules
     • Sample spaces, events and probability axioms
     • Addition and multiplication rules
     • Independence and mutually exclusive events
   • Conditional probability
     • Conditional probability and the multiplication rule
     • Bayes' theorem and updating beliefs
     • Common base-rate fallacies
   • Random variables and expectation
     • Discrete and continuous random variables
     • Expectation, variance and standard deviation
     • Covariance and correlation as moments
2. 2

   Probability Distributions

   3 topics

   • Discrete distributions
     • Bernoulli and binomial distributions
     • Poisson distribution and rare events
   • Continuous distributions
     • The normal distribution and standardisation (z-scores)
     • The exponential and uniform distributions
     • t, chi-squared and F distributions and their uses
   • Sampling distributions
     • Distribution of the sample mean
     • The Central Limit Theorem and its conditions
     • Standard error versus standard deviation
3. 3

   Estimation

   3 topics

   • Point estimation
     • Bias, efficiency and consistency of estimators
     • Method of moments and maximum likelihood (conceptual)
   • Confidence intervals
     • Constructing intervals for means and proportions
     • Interpreting confidence levels correctly
     • Factors affecting interval width
   • Precision and sample size
     • Margin of error and required sample size
     • Trade-offs between precision, cost and effort
4. 4

   Hypothesis Testing

   3 topics

   • Framework of significance testing
     • Null and alternative hypotheses
     • Type I and Type II errors and statistical power
     • Significance level, p-values and their correct interpretation
   • Common tests
     • z-tests and t-tests for means
     • Chi-squared tests for association and goodness of fit
     • Tests for proportions and ANOVA (overview)
   • Pitfalls in inference
     • Multiple comparisons and p-hacking
     • Statistical versus practical significance
     • Misinterpretation of p-values and confidence intervals

Probability and Statistical Inference flashcards for Statistical Officer / Government Statistical Service (GSS) Assessment

18 of 54 cards from the Probability and Statistical Inference deck — real questions with worked answers.

1. State the three axioms of probability (Kolmogorov's axioms).

   For any event $A$: (1) $P(A) \geq 0$; (2) $P(\Omega) = 1$ for the sample space $\Omega$; (3) for mutually exclusive events $A_1, A_2, \ldots$, $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$.

2. What is the addition rule for the probability of the union of two events $A$ and $B$?

   $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. If $A$ and $B$ are mutually exclusive, $P(A \cap B) = 0$ so $P(A \cup B) = P(A) + P(B)$.

3. What is the complement rule in probability?

   $P(A^{c}) = 1 - P(A)$, where $A^{c}$ is the event that $A$ does not occur.

4. Define what it means for two events $A$ and $B$ to be statistically independent.

   $A$ and $B$ are independent if $P(A \cap B) = P(A)\,P(B)$. Equivalently, $P(A \mid B) = P(A)$ (provided $P(B) > 0$).

5. Give the definition of conditional probability $P(A \mid B)$.

   $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, defined for $P(B) > 0$.

6. State the general multiplication rule for $P(A \cap B)$.

   $P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)$.

7. State Bayes' theorem for events $A$ and $B$.

   $P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}$, where typically $P(B) = \sum_i P(B \mid A_i)\,P(A_i)$ via the law of total probability.

8. State the law of total probability for a partition $\{A_1, \ldots, A_n\}$ of the sample space.

   $P(B) = \sum_{i=1}^{n} P(B \mid A_i)\,P(A_i)$, where the $A_i$ are mutually exclusive and exhaustive.

9. Distinguish a discrete random variable from a continuous random variable.

   A discrete random variable takes countably many values (described by a probability mass function $p(x)$); a continuous random variable takes values over an interval and is described by a probability density function $f(x)$, where $P(X = x) = 0$ for any single point.

10. Define the expectation (mean) of a discrete random variable $X$.

    $E[X] = \sum_{x} x\, p(x)$, summing over all values $x$ in the support of $X$.

11. Define the expectation of a continuous random variable $X$ with density $f(x)$.

    $E[X] = \int_{-\infty}^{\infty} x\, f(x)\, dx$.

12. Give the formula for the variance of a random variable $X$ in terms of expectations.

    $\operatorname{Var}(X) = E[(X - \mu)^{2}] = E[X^{2}] - (E[X])^{2}$, where $\mu = E[X]$.

13. State the linearity of expectation property.

    For constants $a, b$ and random variables $X, Y$: $E[aX + bY + c] = a\,E[X] + b\,E[Y] + c$. This holds whether or not $X$ and $Y$ are independent.

14. How does variance behave under a linear transformation $aX + b$?

    $\operatorname{Var}(aX + b) = a^{2}\operatorname{Var}(X)$. Adding a constant $b$ does not change the variance.

15. For independent random variables $X$ and $Y$, what is $\operatorname{Var}(X + Y)$?

    $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$ when $X$ and $Y$ are independent. In general, $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X, Y)$.

16. State the mean and variance of a Bernoulli random variable with success probability $p$.

    $E[X] = p$ and $\operatorname{Var}(X) = p(1 - p)$.

17. Give the probability mass function, mean, and variance of the Binomial distribution $X \sim \text{Bin}(n, p)$.

    $P(X = k) = \binom{n}{k} p^{k}(1 - p)^{n-k}$ for $k = 0, \ldots, n$; mean $E[X] = np$; variance $\operatorname{Var}(X) = np(1 - p)$.

18. Give the probability mass function, mean, and variance of the Poisson distribution with rate $\lambda$.

    $P(X = k) = \dfrac{e^{-\lambda} \lambda^{k}}{k!}$ for $k = 0, 1, 2, \ldots$; mean $E[X] = \lambda$; variance $\operatorname{Var}(X) = \lambda$ (mean equals variance).

See more Probability and Statistical Inference flashcards →

Planning Probability and Statistical Inference for Statistical Officer / Government Statistical Service (GSS) Assessment

Probability and Statistical Inference (Statistical Officer / Government Statistical Service (GSS) Assessment) FAQ