GATE ST: Probability and Distributions

GATE ST treats probability as the foundation for inference, so questions mix axiomatic probability with distributional calculation. Expect direct computation, transformations, conditional reasoning, and limiting theorems rather than descriptive definitions.

Core rules

• Probability is countably additive on disjoint events; use complements and inclusion–exclusion to avoid double counting.
• Conditional probability gives Bayes' rule: P(A|B)=P(B|A)P(A)/P(B). Independence means P(A∩B)=P(A)P(B), which is stronger than zero covariance.
• For a transformation Y=g(X), use the Jacobian when g is one-to-one; otherwise sum densities over preimages. Check the support before integrating.
• MGFs and characteristic functions identify distributions when they exist in a neighborhood of zero. Moments may not determine a distribution without extra conditions.
• Covariance measures linear association only. Var(aX+bY)=a²Var(X)+b²Var(Y)+2abCov(X,Y).
• The weak law concerns convergence in probability; the CLT gives √n(X̄−μ) ⇒ N(0,σ²) under finite variance.

Worked example

Let X∼Exp(λ). Then P(X>s+t|X>s)=P(X>t) because P(X>s+t)/P(X>s)=e^{−λ(s+t)}/e^{−λs}=e^{−λt}. This memoryless property is unique to the exponential among continuous distributions. For a sum of n i.i.d. exponentials, use the gamma distribution, not the CLT, when an exact probability is requested.

Traps

Do not confuse f(x) with P(X=x) for continuous X; a density can exceed 1. When using the CLT, state whether you are approximating a sum or a mean and apply the continuity correction for lattice variables. For distributions with heavy tails, the CLT may fail because the variance is infinite.