UGC NET Economics June 2025 Unit 3: Statistics and Econometrics PYQs (English–Hindi)

Unrelated nonstationary series can yield a high R² and apparently significant coefficients while residuals show strong positive serial correlation. The warning rule is R² > d, not a formal proof of spuriousness.
असंबंधित nonstationary series में high R² और apparently significant coefficients मिल सकते हैं, जबकि residuals में strong positive serial correlation होती है। Warning rule R² > d है; यह spuriousness का formal proof नहीं है।

Example: R² = 0.95 and d = 0.20 warrant checking stationarity and cointegration before trusting the regression. A high R² alone does not validate a meaningful relation.
Example: R² = 0.95 और d = 0.20 होने पर regression पर भरोसा करने से पहले stationarity और cointegration जाँचें। केवल high R² meaningful relation साबित नहीं करता।

Formula: Complement rule with independent events. Use: Subtract the chance of rejection in all three selections from one.
Formula: Independent events का complement rule। उपयोग: तीनों selections में rejection की probability को one से घटाना।

The calculation assumes one selection per post, equal chances within each interview and independence across selections. Without an independence assumption, the three marginal probabilities alone are insufficient.
Calculation में हर post पर एक selection, candidates के equal chances और तीनों selections की independence assumed है। Independence के बिना केवल तीन marginal probabilities पर्याप्त नहीं हैं।

Formula: Joint probability by counting outcomes. Use: Count equally likely ordered dice outcomes satisfying both events.
Formula: Outcomes गिनकर joint probability। उपयोग: दोनों events satisfy करने वाले equally likely ordered dice outcomes गिनना।

A and C are pairwise independent: P(A)P(C) = (1/2)(1/2) = 1/4. The fact that a conditional calculation is possible does not make them dependent.
A और C pairwise independent हैं: P(A)P(C) = (1/2)(1/2) = 1/4। Conditional calculation करना संभव होने से events dependent नहीं हो जाते।

Both dice must be odd. Each die has 3 odd faces out of 6 and the throws are independent, so multiply the two probabilities.
दोनों dice odd होने चाहिए। हर die पर 6 में 3 odd faces हैं और throws independent हैं, इसलिए probabilities multiply करें।

Formula: Product rule for independent events. Use: Find the probability that both independent events occur.
Formula: Independent events का product rule। उपयोग: दो independent events के एक साथ होने की probability निकालना।

If the second die is odd and the sum is odd, the first die must be even. There are 3 × 3 = 9 favourable ordered outcomes out of 36, giving 1/4.
Second die odd हो और sum odd चाहिए तो first die even होना चाहिए। कुल 36 में 3 × 3 = 9 favourable ordered outcomes हैं, इसलिए probability 1/4 है।

B and C are also pairwise independent because their joint probability equals (1/2)(1/2). Do not label them dependent just because the verbal events mention the same dice.
B और C भी pairwise independent हैं क्योंकि joint probability (1/2)(1/2) है। Verbal events में वही dice आने से वे dependent नहीं हो जाते।

Formula: Addition rule for disjoint cases. Use: Add the mutually exclusive odd even and even odd possibilities.
Formula: Disjoint cases का addition rule। उपयोग: Odd even और even odd के mutually exclusive cases जोड़ना।

There are 18 odd sum outcomes among 36 equally likely ordered outcomes. Odd plus odd is even, not odd.
36 equally likely ordered outcomes में 18 का sum odd है। Odd plus odd का sum even होता है, odd नहीं।

A and B together require both dice to be odd, whose sum is even. That cannot satisfy C. The intersection is empty, so its probability is zero.
A और B में दोनों dice odd होने चाहिए, जिनका sum even है। यह C को satisfy नहीं करता। Intersection empty है, इसलिए probability zero है।

High yield: A B C are pairwise independent but not mutually independent. Each pair has probability 1/4, while the triple intersection is 0 rather than (1/2)³ = 1/8. Do not add the three marginal probabilities to obtain the intersection.
High yield: A B C pairwise independent हैं लेकिन mutually independent नहीं। हर pair की probability 1/4 है, पर triple intersection 0 है, (1/2)³ = 1/8 नहीं। Intersection निकालने के लिए marginal probabilities न जोड़ें।

For a chi square variable with n degrees of freedom, kurtosis β₂ = 3 + 12/n, skewness = √(8/n), the mode is n − 2 for n ≥ 2, the third central moment is 8n, and the fourth central moment is 12n² + 48n.
n degrees of freedom वाली chi square variable में kurtosis β₂ = 3 + 12/n, skewness = √(8/n), n ≥ 2 पर interior mode n − 2, third central moment 8n और fourth central moment 12n² + 48n होता है।

Printed IV is √(2/n), missing a factor 2 relative to the usual skewness. Printed D asks for μ₃, but II gives μ₄. Thus (d) is only the likely intended matching after both corrections; it is not correct for the literal printed list.
Printed IV में √(2/n) है, जिसमें usual skewness के मुकाबले factor 2 छूटा है। D में μ₃ पूछा है लेकिन II में μ₄ का formula है। इसलिए दोनों corrections के बाद (d) likely intended matching है; literal printed list के लिए सही नहीं।

Check at n = 8: correct skewness is 1, whereas printed √(2/8) equals 0.5. Keep kurtosis and excess kurtosis separate.
n = 8 पर check करें: correct skewness 1 है, लेकिन printed √(2/8) = 0.5 है। Kurtosis और excess kurtosis को अलग रखें।

Formula: MGF identification for binomial and Poisson distributions. Use: Compare a given MGF with the standard functional form.
Formula: Binomial और Poisson की MGF पहचान। उपयोग: Given MGF को standard form से compare करना।

A is binomial with n = 6 and p = 1/2, not Poisson. B is binomial with n = 5 and p = 1/2. C is Poisson with λ = 3 and D is Poisson with λ = 1/4.
A में n = 6, p = 1/2 वाला binomial है, Poisson नहीं। B में n = 5, p = 1/2 वाला binomial है। C में λ = 3 और D में λ = 1/4 वाला Poisson है।

E simplifies to [(2/3)eᵗ/(1 − (1/3)eᵗ)]², an MGF for the number of trials to the second success with success probability 2/3. It is negative binomial, not binomial. For a quick check, a valid MGF has M(0) = 1.
E को simplify करने पर [(2/3)eᵗ/(1 − (1/3)eᵗ)]² मिलता है। यह success probability 2/3 पर second success तक trials की negative binomial MGF है, binomial नहीं। Quick check में valid MGF के लिए M(0) = 1 होता है।

Formula: Poisson probability recurrence. Use: Obtain the next probability from the current one without recalculating the full PMF.
Formula: Poisson probability recurrence। उपयोग: Full PMF फिर निकालने के बजाय current probability से next probability निकालना।

Divide the PMF at x + 1 by the PMF at x; common factors cancel and (x + 1)! = (x + 1)x!. Example: λ = 3 and x = 2 give P(3)/P(2) = 1, so those probabilities are equal.
x + 1 की PMF को x की PMF से divide करें; common terms cancel और (x + 1)! = (x + 1)x! होता है। Example: λ = 3 और x = 2 में P(3)/P(2) = 1 है; दोनों probabilities equal हैं।

A III and D I are standard: the hypergeometric mean is nM/N and geometric variance is q/p² for success probability p. For example N = 100, M = 20 and n = 10 give a hypergeometric mean of 2. With p = q = 1/2 geometric variance is 2.
A III और D I standard हैं: hypergeometric mean nM/N और success probability p पर geometric variance q/p² है। Example: N = 100, M = 20, n = 10 में mean 2; p = q = 1/2 में geometric variance 2 है।

IV is binomial excess kurtosis, not ordinary kurtosis β₂. Ordinary β₂ is 3 + (1 − 6pq)/(npq). Printed II can be a negative binomial MGF under an appropriate scaled P Q parametrisation with Q − P = 1; it is not the standard formula if P and Q are simply probabilities summing to 1.
IV binomial का excess kurtosis है, ordinary kurtosis β₂ नहीं। Ordinary β₂ = 3 + (1 − 6pq)/(npq) है। Printed II appropriate scaled P Q parametrisation में negative binomial MGF हो सकती है, जहाँ Q − P = 1; P और Q को sum one वाली probabilities मानने पर यह standard formula नहीं है।

Thus the intended option is (b), but do not memorise IV as ordinary kurtosis or use II without checking its parameter definitions.
इसलिए intended option (b) है; IV को ordinary kurtosis या II को parameter definitions देखे बिना याद न करें।

A and E are exact. A normal distribution is symmetric, so μ₃ = 0; its characteristic function is exp(iμt − σ²t²/2). B is false because ordinary β₂ = 3, while excess kurtosis is zero. D is false because μ₄ = 3σ⁴.
A और E exact हैं। Normal distribution symmetric होने से μ₃ = 0 और characteristic function exp(iμt − σ²t²/2) है। B गलत है: ordinary β₂ = 3, जबकि excess kurtosis zero है। D गलत है क्योंकि μ₄ = 3σ⁴ है।

Formula: Mean absolute deviation of a normal variable. Use: Distinguish the exact mean deviation from the common 0.8σ shortcut.
Formula: Normal variable का mean absolute deviation। उपयोग: Exact mean deviation और common 0.8σ shortcut में अंतर पहचानना।

For σ = 5 the exact mean deviation is about 3.9894, not exactly 4. Printed C uses 4σ/5 = 0.8σ as a rounded rule. If equality is interpreted strictly, only A E are true and no option lists that pair; (a) is the intended exam approximation.
σ = 5 पर exact mean deviation लगभग 3.9894 है, exactly 4 नहीं। Printed C में 4σ/5 = 0.8σ rounded rule है। Equality को strict मानें तो केवल A E सही हैं, जिनका option नहीं है; (a) intended exam approximation है।

Formula: Sample size for estimating a population mean. Use: Use the estimated standard deviation and required margin of error.
Formula: Population mean estimation की sample size। उपयोग: Estimated standard deviation और required margin of error से sample size निकालना।

The question’s wording omits what the 16 minute estimate measures. Option (d) requires treating it as σ or an estimated standard deviation. Round the required sample size upward, not to 245.
Question में स्पष्ट नहीं है कि 16 minutes किस quantity का estimate है। Option (d) के लिए इसे σ या estimated standard deviation मानना जरूरी है। Required sample size को ऊपर round करें; 245 न लें।

The nonparametric bootstrap repeatedly resamples the observed sample with replacement, recomputes a statistic and uses its empirical distribution to estimate uncertainty. It can provide standard errors and confidence intervals without requiring a closed form sampling distribution.
Nonparametric bootstrap observed sample से बार बार replacement के साथ resampling करता है, statistic फिर निकालता है और उसकी empirical distribution से uncertainty estimate करता है। इससे standard errors और confidence intervals मिल सकते हैं।

Example: resample five observed incomes many times, calculate the mean each time, and examine the spread of those means. For dependent time series, ordinary independent resampling may be inappropriate; a suitable block method may be needed.
Example: पाँच observed incomes से कई बार resample करके हर बार mean निकालें और इन means का spread देखें। Dependent time series में ordinary independent resampling उचित नहीं हो सकती; suitable block method चाहिए।

ILS first estimates the reduced form by OLS and then recovers structural coefficients through the relationships between reduced form and structural parameters. The target is an exactly identified structural equation.
ILS पहले reduced form को OLS से estimate करता है, फिर reduced form और structural parameters के relations से structural coefficients निकालता है। Target exactly identified structural equation है।

Do not confuse an intermediate estimation step with the final target. Underidentification prevents unique recovery; overidentified equations generally require methods such as 2SLS rather than simple ILS inversion.
Intermediate step और final target अलग रखें। Underidentification में unique recovery नहीं होती; overidentified equations के लिए simple ILS inversion के बजाय सामान्यतः 2SLS जैसी methods चाहिए।

Formula: Likelihood ratio test for joint significance. Use: Compare the intercept only model with the model containing the slope coefficients.
Formula: Joint significance का likelihood ratio test। उपयोग: Intercept only model और slope coefficients वाले model को compare करना।

Under the null and regularity conditions, LR is asymptotically χ² with degrees of freedom equal to the number of restrictions. Example: log likelihoods −90 and −100 give LR = 20. The statistic is LR; χ² describes its reference distribution.
Null और regularity conditions में LR asymptotically χ² होता है; degrees of freedom restrictions की संख्या है। Example: log likelihoods −90 और −100 से LR = 20 है। Statistic LR है; χ² उसकी reference distribution है।

Formula: Regression line of y on x. Use: Predict y from x using the y on x slope and the two sample means.
Formula: y on x की regression line। उपयोग: y on x slope और sample means से x के आधार पर y predict करना।

Check: substituting x = 32 gives y = 38, as a regression line with an intercept passes through the sample means. Use byx, not bxy, when y is the dependent variable.
Check: x = 32 रखने पर y = 38 आता है; intercept वाली regression line sample means से गुजरती है। y dependent variable हो तो byx लें, bxy नहीं।

Formula: Regression prediction using correlation. Use: Use the correlation and standard deviations to obtain the sales on advertisement slope.
Formula: Correlation से regression prediction। उपयोग: Correlation और standard deviations से sales on advertisement slope निकालना।

The units are crore rupees. The estimate is a regression prediction; the correlation alone does not prove that advertising causes the full predicted change in sales.
Units crore rupees हैं। यह regression prediction है; केवल correlation से यह साबित नहीं होता कि sales का पूरा predicted change advertising के कारण ही है।