Statistics for IIT JAM MS
Statistics
The Statistics half tests probability modelling and the logic of inference: standard distributions, expectation and moments, joint and conditional behaviour, sampling distributions, estimation theory, and hypothesis testing. Marks are lost mostly to sign, divisor and support errors rather than to unfamiliar theory.
Probability and distributions
- Bayes' theorem, P(A|B)=P(B|A)P(A)/P(B), is computed through the total probability denominator over a partition.
- Independence means P(A∩B)=P(A)P(B); pairwise independence does not imply mutual independence, and Cov(X,Y)=0 does not imply independence.
- Memorylessness holds only for the geometric and exponential laws.
- For transformations use the CDF method, or the Jacobian f_Y(y)=f_X(x)|dx/dy| in the monotone case.
Inference
- For a normal sample, (n−1)S²/σ² ~ χ²_{n−1} and (X̄−μ)/(S/√n) ~ t_{n−1}; two variances are compared by an F ratio.
- MSE = Var(θ̂) + bias(θ̂)²; sufficiency is checked by the factorization theorem.
- Neyman–Pearson: the most powerful test rejects for large values of the likelihood ratio, with size P(reject | H₀) fixed at α.
Worked example
Let f(x;θ)=θx^(θ−1) on 0<x<1. Then ℓ = n ln θ + (θ−1)Σ ln x_i, so ℓ′ = n/θ + Σ ln x_i = 0 gives θ̂ = −n/Σ ln x_i. Since ℓ″ = −n/θ² < 0, this is a maximum; and because each ln x_i < 0, θ̂ > 0 is admissible.
Traps
Calling ρ=0 independence; writing Var(X̄)=σ² instead of σ²/n; using z-critical values when σ is estimated by S; and MLEs that sit at the boundary of the support - for Uniform(0,θ) the MLE is max x_i, not the sample mean.
Key points
- Bayes problems need the full partition in the denominator.
- Covariance zero never proves independence.
- χ² and t sampling distributions presuppose normal populations.
- MSE decomposes into variance plus squared bias.
- Check whether an MLE lies in the parameter space and on the support boundary.