(a) Let probability of obtaining Head on a biased coin be 4/5 and X be the number of heads obtained in a sequence of 25 independent tosses of the coin. The same coin is tossed again X number of times independently and we obtain Y heads. Compute Var.(

Question

(a) Let probability of obtaining Head on a biased coin be 4/5 and X be the number of heads obtained in a sequence of 25 independent tosses of the coin. The same coin is tossed again X number of times independently and we obtain Y heads. Compute Var.(X+25Y). (20 marks)

(b)(i) Let {6, –8, 3, 2, 7, 5, 4, 9} be a random sample from a population with probability density function f(x, θ) = ½ exp(–|x–θ|), –∞<x, θ<∞. Obtain maximum likelihood estimate of θ. (5 marks)

(b)(ii) Let X₁, X₂, ..., Xₙ be a random sample from Bernoulli distribution b(1, θ), 0<θ<1. Find the lower bound for the variance of an unbiased estimator of θ based on this data. Find uniformly minimum variance unbiased estimator of θ and show that it attains Cramer-Rao lower bound. (10 marks)

(c) Let X₁, X₂, ..., Xₙ be a random sample from beta distribution of first kind β₍₁, θ₎, θ>0. Find consistent estimator of θ, and its variance also. (15 marks)

UPSC Answer Check · Accepted Answer

Calculate the required quantities systematically across all three parts. For part (a), identify distributions and apply variance decomposition for compound random variables; for (b)(i), derive the MLE using the Laplace distribution's median property; for (b)(ii), establish the Cramer-Rao bound and verify attainment; for (c), use method of moments or MLE for consistency. Allocate approximately 40% time to part (a) given its 20 marks, 30% to part (c) for 15 marks, and 30% to part (b) combining 5+10 marks. Present derivations stepwise with clear notation before substituting numerical values.
- Part (a): Correctly identify X ~ Binomial(25, 4/5) and Y|X=x ~ Binomial(x, 4/5), then apply law of total variance to find Var(X+25Y) using E[Var(Y|X)] + Var(E[Y|X]) components
- Part (b)(i): Recognize f(x,θ) as Laplace distribution with location parameter θ, hence MLE of θ equals the sample median (4.5 or between 4 and 5)
- Part (b)(ii): Derive Cramer-Rao lower bound as θ(1-θ)/n, identify sample mean as UMVUE, and prove it attains the bound by showing equality in Cauchy-Schwarz
- Part (c): For Beta(1,θ), derive method of moments estimator θ̂ = (1-X̄)/X̄ or MLE, prove consistency via weak law of large numbers, and compute asymptotic variance
- Correct application of variance formulas: Var(X) = np(1-p) for binomial, and careful handling of the 25Y scaling factor in part (a)
- Proper justification of why sample mean is UMVUE in (b)(ii) using completeness and sufficiency of T = ΣXi, or direct variance calculation
- Verification that estimator in (c) is consistent by showing plim(θ̂) = θ as n → ∞, with explicit variance expression involving θ and n

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies all distributions: X~Bin(25,4/5), Y\|X~Bin(X,4/5) in (a); Laplace with median MLE in (b)(i); Bernoulli with Fisher information in (b)(ii); Beta(1,θ) with E[X]=1/(1+θ) in (c). Properly states all assumptions and parameter spaces.	Identifies most distributions correctly but may confuse conditional distribution in (a) or misstate Beta mean in (c); minor errors in parameter specification.	Major errors in distribution identification, such as treating Y as independent of X in (a), or confusing Beta(1,θ) with Beta(θ,1); fundamentally wrong setup.
Method choice	20%	10	Selects optimal methods: law of total variance for compound variance in (a); sample median for Laplace MLE via absolute value minimization in (b)(i); Cramer-Rao inequality with regularity conditions check in (b)(ii); method of moments/MLE with consistency proof in (c).	Uses correct general approaches but may miss efficiency in (b)(ii) or use brute force where elegant methods exist; attempts correct methods with some procedural gaps.	Wrong methods entirely, such as direct independence assumption for X+25Y in (a), or sample mean for Laplace MLE in (b)(i); no awareness of conditional structure.
Computation accuracy	25%	12.5	Flawless calculations: Var(X)=4, E[X]=20, E[Y]=16, Var(Y\|X) terms, final Var(X+25Y)=2504; MLE=4.5 or interval [4,5]; CRLB=θ(1-θ)/n with verification; consistent estimator θ̂=(1-X̄)/X̄ with correct variance derivation.	Correct methodology but arithmetic slips in variance expansion, especially handling the 25² coefficient; correct MLE identification but wrong numerical value; partial CRLB derivation.	Major computational errors, incorrect variance formula application, wrong scaling (25 vs 25²), or completely wrong final answers despite correct setup.
Interpretation	20%	10	Explains why variance is large in (a) due to compounding; justifies median over mean for Laplace; clearly shows why sample mean attains bound via score function proportionality in (b)(ii); explains consistency intuitively and proves formally using WLLN in (c).	Provides some interpretation but lacks depth on why specific estimators work; mentions consistency without proper limit argument; partial explanation of CRLB attainment.	No interpretation of results, purely mechanical computation; no understanding of why median is MLE for Laplace or why Bernoulli mean achieves bound.
Final answer & units	15%	7.5	All final answers clearly boxed/stated: Var(X+25Y)=2504 (or exact fraction 6260); θ̂_MLE=4.5 or {4,5}; CRLB=θ(1-θ)/n with UMVUE=X̄; consistent estimator θ̂=(1-X̄)/X̄ with asymptotic variance θ(θ+1)²/n. Proper notation throughout.	Most answers present but format inconsistent; one part missing final answer or with wrong notation; incomplete specification of variance in (c).	Missing multiple final answers, confused notation, or answers without context; no clear identification of which result answers which sub-part.

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