India · Reserve Bank of India Services Board · civil-service · unverified source
Phase I objective screening, Phase II papers in statistics and English, and an interview (reported).
Built from Laisha's knowledge because the official page could not be opened or was not found during research (web search quota exhausted); confirm against the body's current notification before relying on it. RBI recruits statisticians for its surveys and econometric work through this stream. The recruitment portal was unreachable during the earlier track research too.
Official source: https://opportunities.rbi.org.in/
Original questions written to this exam's published syllabus and pattern. Not past papers.
1. Let A be an event with ∅ ≠ A ≠ Ω. What is the smallest σ-field on Ω that contains A?
D. {∅, A, Aᶜ, Ω}
Any σ-field containing A must contain Aᶜ (complements) and Ω and ∅. The collection {∅, A, Aᶜ, Ω} already satisfies all three conditions, so it is the smallest. {∅, A, Ω} lacks Aᶜ; {A, Aᶜ} lacks Ω; the power set contains A but is generally far larger than necessary.
Probability Theory · Sample spaces, sigma-fields and the axioms of probability
2. Which discrete distribution on the non-negative integers has the memoryless property P(X ≥ s + t | X ≥ s) = P(X ≥ t)?
D. The geometric distribution
Only the geometric is memoryless among discrete distributions, just as the exponential is among continuous ones. The Poisson, despite its link with the Poisson process, is not memoryless as a count.
Probability Distributions & Sampling Distributions · Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial
3. What is a sample path of a stochastic process?
C. One realisation over time
Fixing the random outcome ω gives a function of t, the trajectory actually observed, e.g. one season's daily rainfall sequence.
Stochastic Processes · Definition and classification of stochastic processes
4. Which estimator of σ² from a normal sample is consistent but biased?
C. Σ(Xᵢ − X̄)²/n
The divisor-n estimator has expectation (n − 1)σ²/n → σ² and variance → 0, so it is consistent but biased. s² is unbiased and consistent; (X₁ − X₂)²/2 is unbiased but not consistent, since it uses only two observations.
Statistical Inference I: Estimation · Point estimation: unbiasedness, consistency and efficiency
5. What is the power of a test at a particular alternative θ₁?
B. The probability of rejecting H₀ when θ₁ is the true value
Power = 1 − β(θ₁) = P_θ₁(reject H₀). Accepting H₀ under θ₁ is the Type II error probability β; rejecting a true H₀ has probability α. The posterior probability of H₁ is a Bayesian quantity, not power.
Statistical Inference II: Testing of Hypotheses & Sequential Analysis · Hypotheses, the two types of error, size and power
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