Probability and Mathematical Statistics
Probability and Mathematical Statistics
This domain tests the rules of probability, conditioning, distributional reasoning, and the construction of estimators and tests. Exam items often require choosing the correct probability statement before any computation.
Core rules and concepts
- Probability axioms: P(A∪B)=P(A)+P(B)−P(A∩B); independence gives P(A∩B)=P(A)P(B).
- Conditional probability: P(A|B)=P(A∩B)/P(B); Bayes reverses the conditioning.
- The law of total probability partitions the sample space.
- Expectation is linear: E[aX+bY]=aE[X]+bE[Y]; Var(aX+bY)=a²Var(X)+b²Var(Y)+2abCov(X,Y).
- Standard distributions: Bernoulli, binomial, Poisson, geometric, uniform, exponential, normal; know their means and variances.
- MLE maximizes the likelihood or log-likelihood; method of moments equates sample and population moments.
- CLT: standardized sums of iid variables approach N(0,1); it applies to means, not individual observations.
Worked example
A disease has prevalence 1%. A test has sensitivity 0.95 and specificity 0.90. Given a positive test, P(D|+)=0.95×0.01/[0.95×0.01+0.10×0.99]=0.0095/0.1085≈0.0876. The low result follows from the much larger healthy group producing false positives.
Traps that cost marks
- Confusing P(A|B) with P(B|A).
- Adding variances without covariance for dependent variables.
- Using a normal approximation without checking np and n(1−p).
- Treating an MLE as automatically unbiased.
- Forgetting that P(positive)=sensitivity×prevalence+(1−specificity)×(1−prevalence).