Probability and Statistics
Probability and Statistics in GATE MA tests distributional calculations, limit theorems, estimation, and hypothesis testing. Problems often require deriving an MLE or applying the CLT with a continuity correction.
Core rules
- For a random variable X, E[X]=∫x f(x)dx or Σx p(x); Var(X)=E[X²]−(E[X])².
- Binomial(n,p): mean np, variance np(1−p). Poisson(λ): mean λ, variance λ. Exponential(λ): mean 1/λ, variance 1/λ².
- CLT: (X̄−μ)/(σ/√n) → N(0,1) in distribution as n→∞.
- MLE maximizes the likelihood L(θ); often maximize log L.
- A test at level α rejects H_0 when the p-value ≤ α; the p-value is not P(H_0 true).
- MGF M_X(t)=E[e^{tX}]; for normal, M_X(t)=e^{μt+σ²t²/2}.
Worked example
Let X_1,…,X_n be iid Exponential(λ) with density λe^{−λx}. The likelihood is L(λ)=λ^n e^{−λΣx_i}. Then log L=n log λ−λΣx_i, and d/dλ=n/λ−Σx_i=0 gives λ̂=n/Σx_i=1/X̄. The second derivative is negative, so this is the MLE.
Traps
- Independence is stronger than zero correlation; uncorrelated normal variables are independent, but not in general.
- The CLT concerns the distribution of the sample mean, not the raw data; sample size matters for accuracy.
- An unbiased estimator is not automatically best; compare variance (efficiency).
- A confidence interval is a random interval, not a probability statement about a fixed parameter after the data are observed.