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

$$E(X^n) = \frac{1}{5} + (-1)^n \frac{2}{5} + \frac{2^{n+1}}{5}; \quad n = 1, 2, 3, \cdots$$

then find the values of the following:

(i) $P(|X - 0.75|

Question

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

$$E(X^n) = \frac{1}{5} + (-1)^n \frac{2}{5} + \frac{2^{n+1}}{5}; \quad n = 1, 2, 3, \cdots$$

then find the values of the following:

(i) $P(|X - 0.75| \leq 1.5)$ (10 marks)

(ii) $P(|X - \mu| < \sigma)$; $\mu = E(X)$ and $\sigma^2 = 	ext{var}(X)$ (10 marks)

[Use $\sqrt{1.84} = 1.36$]

(b) (i) Write the importance of Cramer-Rao inequality and Rao-Blackwell theorem. (5 marks)

(ii) Let $X \sim B(1, 	heta)$, then find the uniformly minimum variance unbiased estimator (UMVUE) of $	heta(1-	heta)$. (10 marks)

(c) Obtain the maximum likelihood estimates of $\alpha$ and $\beta$ for a random sample from the exponential population

$$f(x; \alpha, \beta) = Ce^{-\beta(x-\alpha)}, \alpha \leq x < \infty, \beta > 0$$ (15 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by first identifying the probability distribution from the given moment pattern in part (a), then applying appropriate estimation theory for parts (b) and (c). Allocate approximately 35% time to part (a) (20 marks), 25% to part (b) (15 marks), and 40% to part (c) (15 marks) based on computational complexity. Structure as: distribution identification → probability calculations → theoretical exposition → UMVUE derivation → MLE derivation with likelihood analysis.
- For (a): Identify X as a discrete mixture distribution with P(X=-1)=2/5, P(X=0)=1/5, P(X=2)=2/5 by comparing E(X^n) with MGF expansion or direct pattern recognition from the given moment formula
- For (a)(i): Calculate P(|X-0.75|≤1.5) = P(-0.75≤X≤2.25) by enumerating which mixture components satisfy the inequality, yielding P(X=-1)+P(X=0)+P(X=2)=1 or appropriate subset
- For (a)(ii): Compute μ=E(X)=0.6 and σ²=Var(X)=1.84, then find P(|X-0.6|<1.36) using the identified distribution support points
- For (b)(i): Explain Cramer-Rao inequality provides variance lower bound for unbiased estimators enabling efficiency comparison; Rao-Blackwell theorem enables improvement of unbiased estimators via conditioning on sufficient statistics
- For (b)(ii): Derive UMVUE of θ(1-θ) using Lehmann-Scheffé theorem: identify T=ΣX_i as complete sufficient statistic, find unbiased estimator based on sample variance or direct calculation, condition to obtain final form
- For (c): Obtain MLEs by writing likelihood L(α,β)=C^n exp[-βΣ(x_i-α)] with constraint α≤x_(1), show likelihood increases with α so α̂=X_(1), then maximize with respect to β to get β̂=n/[Σ(X_i-X_(1))]

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	12	Correctly identifies the mixture distribution in (a) with proper justification; correctly states Bernoulli model and sufficient statistic for (b); properly writes likelihood function with parameter constraints for (c)	Identifies distribution in (a) with minor errors; states model for (b) but incomplete; writes likelihood for (c) but misses constraint α≤x_(1)	Fails to identify distribution in (a); incorrect model specification in (b); fundamentally wrong likelihood setup in (c)
Method choice	20%	12	Uses MGF pattern recognition or moment-matching for (a); applies Lehmann-Scheffé theorem correctly for (b)(ii); employs order statistic argument and profile likelihood correctly for (c)	Uses correct general methods but with some procedural gaps; attempts Rao-Blackwell conditioning but incomplete; uses standard differentiation for β but misses boundary solution for α	Uses inappropriate methods (e.g., assumes normality in (a)); attempts direct expectation without conditioning for UMVUE; fails to recognize boundary MLE for α
Computation accuracy	20%	12	Accurate computation of all moments in (a); correct arithmetic for probabilities using √1.84=1.36; precise derivation of UMVUE formula; exact MLE expressions with correct denominators	Minor arithmetic errors in probability calculations; correct method but algebraic slip in UMVUE derivation; correct α̂ but error in β̂ expression	Major computational errors in moments or probabilities; incorrect final UMVUE expression; fundamentally wrong MLE derivation
Interpretation	20%	12	Interprets mixture components meaningfully; explains why P(\|X-μ\|<σ) differs from normal approximation; clearly explains efficiency improvement in (b)(i); justifies why α̂=X_(1) is biased but MLE	Basic interpretation of results without deeper insight; standard statements about theorems without application context; minimal interpretation of MLE properties	No interpretation of probabilistic results; purely mechanical theorem statements; no discussion of estimator properties
Final answer & units	20%	12	All five sub-parts answered with boxed/clear final answers: exact probabilities for (a)(i)-(ii), clear theorem importance summary for (b)(i), explicit UMVUE formula for (b)(ii), and closed-form MLE expressions for (c)	Most answers present but some incomplete or unclear; missing one sub-part answer or poorly formatted final expressions	Missing multiple final answers; answers without supporting work; illegible or disorganized presentation

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