Indian Statistical Institute admission tests Probability Distributions & Sampling Distributions: practice questions

Indian Statistical Institute admission tests (BStat, MStat, JRF in Statistics) · India · Indian Statistical Institute

Original practice questions written to the published syllabus · not past papers

10worked here, free

46written to this exam in this subject

11syllabus units

What this subject covers

Every standard distribution a statistician must recognise and derive from: discrete and continuous families, their relations and limiting forms, compound and truncated distributions, order statistics, the bivariate normal and the sampling distributions chi-square, t and F.

In the exams: AICE-JRF Ag Stat Unit 1 and Unit 2 'compound and truncated distribution, order statistics' (heavy); GATE ST distributions and order statistics (heavy); IIT JAM MS (heavy); UPSC ISS Statistics I (heavy)

Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial
Continuous distributions: uniform, normal, exponential, gamma, beta, Cauchy, Weibull, lognormal
Relations and limiting forms among distributions
Compound, mixture and truncated distributions
Pearsonian system and fitting of distributions to agricultural data
Order statistics: distributions of single, joint and range
Bivariate normal: marginal and conditional distributions
Sampling distributions: chi-square, t and F and their properties
Non-central chi-square, t and F
Exponential family of distributions
Distributions of quadratic forms in normal variables (Cochran's theorem)

Where the 46 questions fall

How our questions for Indian Statistical Institute admission tests spread across the units of Probability Distributions & Sampling Distributions. Difficulty: 17 easy, 25 medium, 4 hard.

Sampling distributions: chi-square, t and F and their properties	8
Continuous distributions: uniform, normal, exponential, gamma, beta, Cauchy, Weibull, lognormal	6
Bivariate normal: marginal and conditional distributions	6
Exponential family of distributions	6
Order statistics: distributions of single, joint and range	5
Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial	4
Relations and limiting forms among distributions	4
Compound, mixture and truncated distributions	3
Distributions of quadratic forms in normal variables (Cochran's theorem)	2
Pearsonian system and fitting of distributions to agricultural data	1
Non-central chi-square, t and F	1

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Continuous distributions: uniform, normal, exponential, gamma, beta, Cauchy, Weibull, lognormal · medium

X has a Cauchy distribution with location 5 and scale 2. What is its interquartile range?

4
1.349 × 2
It does not exist, because no moments exist
2

Answer and working

A. 4

The Cauchy quartiles are θ ± σ = 3 and 7, so the IQR is 2σ = 4. The Cauchy has no mean or variance, but its quantiles exist. 1.349σ is the IQR of a normal.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions

2 · Relations and limiting forms among distributions · easy

V has a chi-square distribution with 2 degrees of freedom. What is P(V > 6)?

0.000006
0.0498
0.0025
0.9502

Answer and working

B. 0.0498

χ²₂ is exponential with mean 2: P(V > 6) = e^(−6/2) = e^(−3) = 0.0498. e^(−6) treats the mean as 1.

3 · Relations and limiting forms among distributions · medium

X₁, ..., X₄ are independent exponential variables with rate 2. What is the distribution of their sum?

Gamma with shape 4 and rate 2, mean 2
Exponential with rate 8, mean 0.125
Exponential with rate 1/2, mean 2
Gamma with shape 2 and rate 4, mean 0.5

Answer and working

A. Gamma with shape 4 and rate 2, mean 2

The MGF of the sum is [2/(2 − t)]⁴, the gamma (Erlang) MGF with shape 4 and rate 2: mean 4/2 = 2. Rate 8 is the distribution of the minimum of the four, not the sum.

4 · Compound, mixture and truncated distributions · medium

Insects per plant are Poisson with mean λ, and λ varies from plant to plant according to a gamma distribution. What is the unconditional distribution of insects per plant?

Poisson with a larger mean
Binomial with n equal to the gamma shape
Gamma with the same shape
Negative binomial

Answer and working

D. Negative binomial

Integrating e^(−λ)λˣ/x! against a gamma(k, β) density gives C(x + k − 1, x)(β/(1 + β))ᵏ(1/(1 + β))ˣ: negative binomial with r = k. The variance exceeds the mean by the variance of the mixing distribution, so it cannot be Poisson.

5 · Order statistics: distributions of single, joint and range · medium

For a random sample of size n from a continuous distribution with CDF F and density f, which is the density of the r-th order statistic X₍ᵣ₎?

n!/[(r − 1)!(n − r)!] F(x)^(n−r) [1 − F(x)]^(r−1) f(x)
C(n, r) F(x)^r [1 − F(x)]^(n−r), with no factor f(x)
n!/[r!(n − r)!] F(x)^r [1 − F(x)]^(n−r) f(x)
n!/[(r − 1)!(n − r)!] F(x)^(r−1) [1 − F(x)]^(n−r) f(x)

Answer and working

D. n!/[(r − 1)!(n − r)!] F(x)^(r−1) [1 − F(x)]^(n−r) f(x)

r − 1 observations fall below x, one at x and n − r above: the multinomial count n!/[(r − 1)! 1! (n − r)!] times F^(r−1) f (1 − F)^(n−r). The C(n, r)Fʳ(1 − F)ⁿ⁻ʳ form is one term of the CDF sum, not a density.

6 · Bivariate normal: marginal and conditional distributions · medium

Plant height X (cm) and grain yield Y (g) are bivariate normal with μ_X = 100, σ_X = 10, μ_Y = 50, σ_Y = 8 and ρ = 0.6. What is the mean yield of plants 110 cm tall?

57.5 g
56.0 g
58.0 g
54.8 g

Answer and working

D. 54.8 g

E(Y | X = x) = μ_Y + ρ(σ_Y/σ_X)(x − μ_X) = 50 + 0.6 × 0.8 × 10 = 54.8 g. Using σ_X/σ_Y inverts the slope and gives 57.5 g; 58.0 g assumes perfect correlation.

7 · Sampling distributions: chi-square, t and F and their properties · medium

A random sample of 16 is drawn from N(μ, 4). What is the variance of the sample variance s² (divisor n − 1)?

1.0667
2.1333
2.0000
30.0000

Answer and working

B. 2.1333

(n − 1)s²/σ² ~ χ²₁₅ with variance 30, so Var(s²) = σ⁴ × 2(n − 1)/(n − 1)² = 2σ⁴/(n − 1) = 32/15 = 2.1333. 30 is the variance of the chi-square itself; dividing by n gives 2.0000.

8 · Sampling distributions: chi-square, t and F and their properties · easy

X has the F distribution with (5, 12) degrees of freedom. What is E(X)?

1.0
1.2
2.4
1.67

Answer and working

B. 1.2

E(F) = n/(n − 2) for denominator df n > 2: 12/10 = 1.2, regardless of the numerator df 5. 1.67 = 5/3 uses the numerator df.

9 · Exponential family of distributions · hard

For a random sample from a beta(a, b) distribution with both parameters unknown, what is the natural sufficient statistic?

(Σ log Xᵢ, Σ log(1 − Xᵢ))
(ΣXᵢ, ΣXᵢ²)
(Π Xᵢ alone)
(ΣXᵢ, Σ(1 − Xᵢ))

Answer and working

A. (Σ log Xᵢ, Σ log(1 − Xᵢ))

log f = (a − 1) log x + (b − 1) log(1 − x) − log B(a, b), so T = (log x, log(1 − x)). Note ΣXᵢ and Σ(1 − Xᵢ) carry the same information and are not sufficient here. ΠXᵢ alone covers only a.

10 · Distributions of quadratic forms in normal variables (Cochran's theorem) · medium

X ~ Nₙ(0, σ²I), A is symmetric and L is a matrix. Under what condition are the linear form LX and the quadratic form X′AX independent?

trace(LA) = n
AL = I
L′L = A
LA = 0

Answer and working

D. LA = 0

LX and X′AX are independent iff LA = 0. With L = (1/n)1′ and A = I − J/n, LA = 0, which proves the independence of X̄ and Σ(Xᵢ − X̄)² for normal samples.

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