IIT JAM Mathematical Statistics: Probability, Inference and Models
IIT JAM MS tests probability, random variables, standard distributions, sampling, estimation, hypothesis testing, regression and experimental design. The exam emphasizes correct conditioning, variance algebra and interpretation of inferential quantities.
Core rules and concepts
- Probability: P(A|B) = P(A ∩ B)/P(B); independence means P(A ∩ B) = P(A)P(B); total probability sums over a partition.
- Random variables: E(aX + bY) = aE(X) + bE(Y); Var(aX + bY) = a²Var(X) + b²Var(Y) + 2abCov(X, Y).
- Distributions: binomial counts successes in fixed independent trials; Poisson models counts; normal approximations rely on the central limit theorem.
- Estimation: unbiasedness is E(θ̂) = θ; consistency concerns convergence in probability; efficiency compares variance among unbiased estimators.
- Testing: Type I error rejects a true null; power = 1 − P(Type II error); the p-value is computed assuming the null is true.
- Regression: OLS minimizes the residual sum of squares; Gauss–Markov assumptions matter for BLUE.
Worked example
A disease has prevalence 1%. A test has sensitivity 99% and specificity 95%. Given a positive test, find P(disease). P(+) = 0.01×0.99 + 0.99×0.05 = 0.0099 + 0.0495 = 0.0594. Then P(D|+) = 0.0099/0.0594 = 1/6 ≈ 0.1667. Despite high sensitivity, the low prevalence keeps the posterior near 17%.
Traps that cost marks
Do not reverse P(A|B) and P(B|A). Sampling without replacement changes independence. Variance of a sum includes covariance unless variables are uncorrelated. A p-value is not the probability that the null hypothesis is true. A frequentist confidence interval does not assign a probability to a fixed parameter.
Key points
- Bayes reverses conditioning; prevalence matters.
- Var(X + Y) includes 2Cov(X, Y).
- Unbiasedness and consistency are different properties.
- Power = 1 − P(Type II error).
- p-value is computed under the null, not as P(H₀ | data).