India · Indian Statistical Institute · pg · unverified source
Written tests in two parts: an objective mathematics/statistics test and a descriptive subject test, followed by interview for some programmes (reported). JRF selection by written test and interview.
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Official source: https://www.isical.ac.in/
Original questions written to this exam's published syllabus and pattern. Not past papers.
1. The set of irrational numbers is:
C. Uncountable
R = Q ∪ irrationals; if the irrationals were countable, R would be a union of two countable sets and hence countable, which it is not.
Real Analysis for Statistics · Sets, countability and the real number system
2. The space of polynomials of degree at most 3 has dimension:
B. 4
A basis is {1, x, x², x³}, so the dimension is 4.
Linear Algebra & Matrix Theory for Statistics · Vector spaces, subspaces, linear independence, basis and dimension
3. A collection 𝓕 of subsets of Ω contains Ω. Which further pair of closure properties makes 𝓕 a σ-field?
B. Closure under complements and under countable unions
A σ-field needs Ω, complements and countable unions; countable intersections then follow by De Morgan. Closure under complements and finite unions only defines a field (algebra), which is weaker. Arbitrary uncountable unions are not required, and without complements the collection need not be a σ-field.
Probability Theory · Sample spaces, sigma-fields and the axioms of probability
4. A geometric variable X counts the failures before the first success in independent trials with success probability 0.25. What is E(X)?
A. 3
For the failures-before-first-success version, E(X) = q/p = 0.75/0.25 = 3. The trials-until-success version has mean 1/p = 4; 12 is the variance q/p².
Probability Distributions & Sampling Distributions · Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial
5. When is a process weakly (covariance) stationary?
B. Constant mean; covariance depends only on lag
Weak stationarity concerns only first and second moments; shift-invariance of all finite-dimensional distributions is strict stationarity. Brownian motion has independent increments but variance growing with t, so it is not stationary.
Stochastic Processes · Definition and classification of stochastic processes
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