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Statistics 2024 Paper I 50 marks Solve

Paper I — Q3

(a) Let moment generating function of random variable X exist in the neighbourhood of zero and if E(Xⁿ) = 1/5 + (-1)ⁿ 2/5 +…

(a)

Let moment generating function of random variable X exist in the neighbourhood of zero and if

E(Xⁿ) = 1/5 + (-1)ⁿ 2/5 + (2ⁿ⁺¹)/5; n = 1, 2, 3, …

then find the values of the following:

(i)

P(|X - 0.75| ≤ 1.5) 10 marks

(ii)

P(|X - μ| < σ); μ = E(X) and σ² = var(X) 10 marks

[Use √1.84 = 1.36]

(b)
(i)

Write the importance of Cramer-Rao inequality and Rao-Blackwell theorem. 5 marks

(ii)

Let X ∼ B(1, θ), then find the uniformly minimum variance unbiased estimator (UMVUE) of θ(1-θ). 10 marks

(c)

Obtain the maximum likelihood estimates of α and β for a random sample from the exponential population

f(x; α, β) = Ce⁻β(x-α), α ≤ x < ∞, β > 0 15 marks

हिंदी में प्रश्न पढ़ें
(a)

मान लीजिए कि, शून्य के सामीप्य में, यादृच्छिक चर X के आघूर्ण जनक फलन का अस्तित्व है और यदि

E(Xⁿ) = 1/5 + (-1)ⁿ 2/5 + (2ⁿ⁺¹)/5; n = 1, 2, 3, …

है, तो निम्नलिखित के मान ज्ञात कीजिए :

(i)

P(|X - 0.75| ≤ 1.5) (10 अंक)

(ii)

P(|X - μ| < σ); μ = E(X) और σ² = var(X) (10 अंक)

[प्रयोग कीजिए √1.84 = 1.36]

(b)
(i)

क्रैमर-राव असमिका एवं राव-ब्लैकवेल प्रमेय के महत्व लिखिए। (5 अंक)

(ii)

मान लीजिए कि X ∼ B(1, θ), तब θ(1-θ) का एकसमान न्यूनतम प्रसरण अनभिनत आकलक (UMVUE) निकालिए। (10 अंक)

(c)

चरघातांकी समिष्टि

f(x; α, β) = Ce⁻β(x-α), α ≤ x < ∞, β > 0

से लिए गए एक यादृच्छिक प्रतिदर्श के लिए α और β के अधिकतम संभाविता आकलक ज्ञात कीजिए। (15 अंक)

Q3 of the 2024 UPSC Mains Statistics Paper I, as printed
The question as printed in the 2024 Statistics paper
Evaluate your answer to this question → See all 2024 Statistics questions

Model answer

Written by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key.

(a)(i) Let M(t)=E(e^(tX)). Since the moment generating function exists in a neighbourhood of 0, M(t)=1+Σ (n=1 to ∞) E(Xⁿ)tⁿ/n!. Substituting the given moments,

M(t)=1+Σ (n=1 to ∞) [1/5+(−1)ⁿ 2/5+2^(n+1)/5]tⁿ/n!

=1+(1/5)(e^t−1)+(2/5)(e^(−t)−1)+(2/5)(e^(2t)−1)

=(1/5)e^t+(2/5)e^(−t)+(2/5)e^(2t).

Thus X has the distribution P(X=−1)=2/5, P(X=1)=1/5, P(X=2)=2/5.

Now |X−0.75|≤1.5 means −0.75≤X≤2.25. Among the support values, X=1 and X=2 satisfy this, while X=−1 does not.

Therefore P(|X−0.75|≤1.5)=P(X=1)+P(X=2)=1/5+2/5=3/5.

Final answer: 3/5.

(a)(ii) From the distribution, μ=E(X)=(−1)(2/5)+(1)(1/5)+(2)(2/5)=3/5=0.6. Also, E(X²)=(1)(2/5)+(1)(1/5)+(4)(2/5)=11/5. Hence σ²=Var(X)=E(X²)−μ²=11/5−(3/5)²=11/5−9/25=46/25=1.84. So σ=√1.84=1.36, as given.

We need P(|X−μ|<σ)=P(|X−0.6|<1.36). This is equivalent to −1.36<X−0.6<1.36, i.e. −0.76<X<1.96. Among X=−1,1,2, only X=1 lies in this interval.

Therefore P(|X−μ|<σ)=P(X=1)=1/5.

Final answer: 1/5.

(b)(i) The Cramer-Rao inequality states that, under regularity conditions, for an unbiased estimator T of φ(θ), Var(T)≥[φ′(θ)]²/I(θ), where I(θ) is the Fisher information. Its importance is that it gives a lower bound on variance, measures efficiency, helps identify UMVUEs, and guides sample-size determination.

The Rao-Blackwell theorem states that if T is a sufficient statistic and U is an unbiased estimator with finite variance, then E(U|T) is also unbiased and Var(E(U|T))≤Var(U). Its importance is that it improves an estimator by conditioning on a sufficient statistic; combined with completeness and the Lehmann-Scheffé theorem, it yields UMVUEs.

(b)(ii) As written, the question says X∼B(1,θ), i.e. a single Bernoulli observation. Any estimator is T(X) with T(0)=a and T(1)=b. Then E[T(X)]=a(1−θ)+bθ=a+(b−a)θ, which is linear in θ, whereas θ(1−θ)=θ−θ² is quadratic. No choice of a and b makes them equal for all θ. Hence no unbiased estimator of θ(1−θ) exists, and therefore no UMVUE exists.

If the intended statement is a random sample X₁,X₂,…,Xₙ∼B(1,θ) with n≥2, let T=ΣXᵢ. Then T is complete sufficient and T∼Bin(n,θ). The estimator h(T)=T(n−T)/(n(n−1)) is unbiased because E[T(n−T)]=n(n−1)θ(1−θ). By the Lehmann-Scheffé theorem, it is the UMVUE.

Final answer: For one observation, no UMVUE exists. For a sample of size n≥2, the UMVUE is T(n−T)/(n(n−1)), where T=ΣXᵢ.

(c) The density is f(x;α,β)=C e^(−β(x−α)), α≤x<∞, β>0. Normalization gives 1=∫ from α to ∞ C e^(−β(x−α)) dx=C∫ from 0 to ∞ e^(−βu) du=C/β, so C=β. Hence f(x;α,β)=β e^(−β(x−α)), α≤x<∞.

For a random sample X₁,…,Xₙ, the likelihood is L(α,β)=βⁿ e^(−βΣ(Xᵢ−α)), provided α≤min Xᵢ; otherwise L=0.

Let X₍₁₎=min(X₁,…,Xₙ). For fixed β, L=βⁿ e^(−βΣXᵢ+nβα). Since β>0, this increases as α increases. The constraint is α≤X₍₁₎. Therefore the maximum occurs at α̂=X₍₁₎.

Substituting α̂=X₍₁₎, ℓ=ln L=n ln β−βΣ(Xᵢ−X₍₁₎). Differentiate: dℓ/dβ=n/β−Σ(Xᵢ−X₍₁₎). Set equal to zero: β̂=n/Σ(Xᵢ−X₍₁₎). Also, d²ℓ/dβ²=−n/β²<0, so this is a maximum. This requires that not all observations are equal; if all observations are equal, the likelihood is unbounded in β.

Final answer: α̂=min Xᵢ, and β̂=n/Σ(Xᵢ−min Xᵢ)=1/(X̄−min Xᵢ), assuming not all observations are equal.

What "Solve" is asking you to do

Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.

Structure that answers it

Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity

Where marks are lost

Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.

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How this answer will be evaluated

Approach

(a(i)) calculate: given > formula > substitution > result with units > interpretation | (a(ii)) calculate: given > formula > substitution > result with units > interpretation | (b(i)) write short notes: define > 3-4 key features > one example > one-line significance | (b(ii)) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: All parts solved with correct method, clear steps, and proper interpretation.

Key points expected

  • Calculate E(X) and E(X²) from the given formula
  • Determine the support of X (values and probabilities)
  • Identify the range of X satisfying the inequality
  • Sum the probabilities of the relevant values
  • Calculate μ = E(X) and σ² = Var(X)
  • Use the provided value √1.84 = 1.36 for σ
  • Determine the range of X satisfying |X - μ| < σ
  • State the Cramer-Rao lower bound for variance

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

  1. (a(i)) Compute the probability P(|X - 0.75| ≤ 1.5) using the given moments. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Calculate E(X) and E(X²) from the given formula
    • Determine the support of X (values and probabilities)
    • Identify the range of X satisfying the inequality
    • Sum the probabilities of the relevant values

    Loses marks

    • Assuming a continuous distribution without justification
    • Incorrect calculation of the support values

    Earns more

    • Explicitly listing the probability mass function
    • Showing the step-by-step substitution for n=1 and n=2

    Extra mark

    • Verifying the sum of probabilities equals 1
  2. (a(ii)) Compute the probability P(|X - μ| < σ) using the calculated mean and variance. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Calculate μ = E(X) and σ² = Var(X)
    • Use the provided value √1.84 = 1.36 for σ
    • Determine the range of X satisfying |X - μ| < σ
    • Sum the probabilities of the relevant values

    Loses marks

    • Using the wrong value for σ (e.g., 1.84 instead of 1.36)
    • Confusing variance with standard deviation

    Earns more

    • Correctly identifying the strict inequality boundary
    • Clear substitution of calculated μ and σ

    Extra mark

    • Comparing the result with the Chebyshev bound
  3. (b(i)) Explain the importance of the Cramer-Rao inequality and Rao-Blackwell theorem. 5 marks

    write short notes— define → 3-4 key features → one example → one-line significance

    Must cover

    • State the Cramer-Rao lower bound for variance
    • Explain Rao-Blackwellization (conditioning on sufficient statistic)
    • Mention the role in finding UMVUE

    Loses marks

    • Confusing the two theorems
    • Failing to mention the reduction in variance

    Earns more

    • Briefly defining the conditions for Cramer-Rao
    • Mentioning the Lehmann-Scheffe theorem connection

    Extra mark

    • Providing a simple example of Rao-Blackwellization
  4. (b(ii)) Find the UMVUE of θ(1-θ) for X ~ B(1, θ). 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Identify a sufficient statistic (e.g., sum of sample)
    • Find an initial unbiased estimator for θ(1-θ)
    • Apply Rao-Blackwell theorem to condition on the sufficient statistic
    • Derive the final UMVUE formula

    Loses marks

    • Skipping the Rao-Blackwell step
    • Incorrectly identifying the sufficient statistic

    Earns more

    • Showing the expectation calculation for the initial estimator
    • Clearly stating the distribution of the sufficient statistic

    Extra mark

    • Verifying the final estimator is unbiased
  5. (c) Obtain the maximum likelihood estimates of α and β for the given exponential population. 15 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Write the likelihood function L(α, β)
    • Write the log-likelihood function l(α, β)
    • Differentiate with respect to β and solve for β
    • Identify the MLE for α (minimum of the sample)

    Loses marks

    • Ignoring the constraint α ≤ x
    • Treating α as a standard parameter for differentiation

    Earns more

    • Correctly handling the indicator function for the support
    • Showing the derivative steps clearly

    Extra mark

    • Discussing the existence of the MLE for α

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