UPHESC Assistant Professor Statistics 2025 PYQs

Concept: Translate each relation into a probability statement.

1. Mutually exclusive: $A\cap B=\phi$ ⇒ $A^C\cup B^C=(A\cap B{)}^C=S$ ⇒ probability 1 (a–III).
2. Independent: $P(A\cap B)=P(A)P(B)$ (b–I).
3. $A\supset B$ ⇒ $A\cup B=A$ ⇒ $P(A\cup B)=P(A)$ (c–IV).
4. $A\subset B$ ⇒ $P(A)\le P(B)$ (d–II).

Short trick: De Morgan: $A^C\cup B^C=(A\cap B{)}^C$.

Concept: Fisher's linear discriminant function $Y={\lambda }^{\prime }x$, $\lambda ={\mathrm{\Sigma }}^{-1}({\mu }_1-{\mu }_2)$, gives a score; the observation is allotted to a class by comparing the score with a cut-off.

1. (A) is true – classification of a new observation from its measured features is exactly the purpose of a discriminant function.
2. (R) is false – the function produces a discriminant score (a linear combination of variables), not a probability of class membership; probabilities come from Bayes' rule or logistic models.
3. Hence (A) true, (R) false.

Short trick: Discriminant function → score and cut-off, not probability.

Concept: Base-year weights overstate and current-year weights understate a price rise.

1. Paasche (current quantities) – downward bias (a–I).
2. Laspeyres (base quantities) – upward bias (b–II).
3. Factor reversal: price index × quantity index = value index $V_{01}$ (c–III).
4. Time reversal: $P_{01}\times P_{10}=1$ (d–IV). Fisher's ideal index satisfies both tests.

Short trick: L-Up, P-Down; FRT = P×Q = V; TRT = P₀₁×P₁₀ = 1.

Concept: $V_{WOR}(\overline{y})=\frac{N-n}{N-1}\cdot \frac{{\sigma }^2}{n}$, $V_{WR}(\overline{y})=\frac{{\sigma }^2}{n}$.

1. Condition: $\frac{N-n}{N-1}=\frac{1}{2}$ ⇒ $2N-2n=N-1$.
2. ⇒ $n=\frac{N+1}{2}$ (the options give the sample size in terms of N).
3. E.g. $N=9$: $n=5$, and $\frac{9-5}{8}=\frac{1}{2}$ ✓.

Short trick: Finite population correction $\frac{N-n}{N-1}=\frac{1}{2}$ ⇒ sample about half the population.

Concept: Sampling error falls as $n$ grows ($V(\overline{y})\propto 1/n$), but non-sampling errors (non-response, measurement, processing) grow with the scale of the survey.

1. (A) is false: a larger sample needs more investigators, more data handling and less supervision per unit, so non-sampling errors tend to increase (they are largest in a census).
2. (R) is true: larger $n$ means smaller standard error, i.e. more precise estimates.
3. Hence (A) false, (R) true.

Short trick: Big n: sampling error ↓, non-sampling error ↑.

Concept: $P=\frac{(\genfrac{}{}{0ex}{}{6}{3})(\genfrac{}{}{0ex}{}{9}{2})}{(\genfrac{}{}{0ex}{}{15}{5})}$.

1. $(\genfrac{}{}{0ex}{}{6}{3})=20$, $(\genfrac{}{}{0ex}{}{9}{2})=36$ ⇒ favourable $=720$.
2. $(\genfrac{}{}{0ex}{}{15}{5})=3003$.
3. $P=720/3003=0.2398\approx 0.239$.

Short trick: Choose from each group separately, divide by choosing from all.

Concept: Every student takes at least one class, so $P(A\cup D\cup S)=1$.

1. $P(A\cap D)=0.50+0.65-0.80=0.35$.
2. $1=0.50+0.65+0.55-0.35-0.25-0.25+P(A\cap D\cap S)$.
3. $1=0.85+P(ADS)$ ⇒ $P(ADS)=0.15$.

Short trick: Union of all = 1 when everyone takes at least one option.

Concept: $r_{12.3}=\frac{r_{12}-r_{13}{r}_{23}}{\sqrt{(1-r_{13}^2)(1-r_{23}^2)}}$.

1. Put all $r=\rho$: $r_{12.3}=\frac{\rho -{\rho }^2}{1-{\rho }^2}$.
2. $=\frac{\rho (1-\rho )}{(1-\rho )(1+\rho )}=\frac{\rho }{1+\rho }$.

Short trick: Equal correlations ρ ⇒ every first-order partial = ρ/(1+ρ).

Concept: Compare $(AB)$ with its expectation $\frac{(A)(B)}{N}$, or use Yule's $Q=\frac{(AB)(\alpha \beta )-(A\beta )(\alpha B)}{(AB)(\alpha \beta )+(A\beta )(\alpha B)}$.

1. $N=1216$, $(A)=304$, $(B)=400$ ⇒ expected $(AB)=\frac{304\times 400}{1216}=100$; observed 256 > 100.
2. $Q=\frac{196608-6912}{196608+6912}=\frac{189696}{203520}\approx0.93>0$.
3. So the attributes are (strongly) positively associated – the precise description.

Short trick: $(AB)(\alpha\beta)>(A\beta)(\alpha B)$ ⇒ positive association.

Concept: $\overline{X}$ chart from ranges: UCL $=\overline{\overline{X}}+A_2\overline{R}$.

1. $\overline{\overline{X}}=525/25=21$, $\overline{R}=90/25=3.6$.
2. UCL $=21+0.729\times 3.6=21+2.624=23.62$ (LCL $=18.38$).
3. $d_2$ is not needed here (it gives $\hat{\sigma }=\overline{R}/d_2$).

Short trick: Mean chart limits: $\overline{\overline{X}}\pm A_2\overline{R}$.

Concept: Direct method: STDR of the study population = $\frac{\sum m_x^b{P}_x^s}{\sum P_x^s}$ – its own age-specific rates applied to the standard population's age structure.

1. Here the standard population (A) is denoted by superscript $q$ and the study population by $b$.
2. STDR $=\frac{\sum m_x^b{P}_x^q}{\sum P_x^q}=\frac{3834000}{122000}=31.43\approx 31.4$.
3. $\frac{\sum m_x^q{P}_x^b}{\sum P_x^b}=33.6$ is only the index (expected) rate used in the indirect method, not the standardized rate.

Short trick: Direct STDR: study rates × standard population ÷ standard total.

Concept: In SRSWOR, $\sum _{i=1}^N(Y_i-\mu )=0$ forces a negative covariance between draws.

1. $\text{Cov}(y_i,y_j)=\frac{1}{N(N-1)}\sum _{a\ne b}(Y_a-\mu )(Y_b-\mu )$.
2. $\sum _{a\ne b}(Y_a-\mu )(Y_b-\mu )={[\sum (Y_a-\mu )]}^2-\sum (Y_a-\mu {)}^2=0-N{\sigma }^2$.
3. So Cov $=-\frac{{\sigma }^2}{N-1}$ (with ${\sigma }^2=\frac{1}{N}\sum (Y_a-\mu {)}^2$).

Short trick: Without replacement: a high first draw makes the next draw lower → covariance negative, $-{\sigma }^2/(N-1)$.

Concept: $P_{01}^F=\sqrt{\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times \frac{\sum p_1{q}_1}{\sum p_0{q}_1}}\times 100$.

1. Laspeyres $=\frac{1000}{650}=1.5385$; Paasche $=\frac{1300}{1500}=0.8667$.
2. Product $=\frac{1300000}{975000}=1.3333$.
3. $\sqrt{1.3333}=1.1547$ ⇒ Fisher index $=115.47$.

Short trick: Fisher = G.M. of Laspeyres and Paasche.

Concept: $X_n\stackrel{a.s.}{\to }X$ ⇒ $X_n\stackrel{P}{\to }X$, but not conversely.

1. a.s. convergence means $P(lim{X}_n=X)=1$; then for every $\epsilon$, $P(\sup_{m\ge n}|X_m-X|>\varepsilon)\to0$,whichgives$P(|X_n-X|>\varepsilon)\to0$.
2. So (A) is true, and (R) states exactly this general implication – it is the reason (A) holds.
3. Counter-example for converse: independent $X_n\sim$ Bernoulli$(1/n)$ converge to 0 in probability but not a.s.

Short trick: Stronger mode always implies the weaker one.

Concept: Hotelling's $T^2=\nu X^{\prime }{T}^{-1}X$ arises from a normal vector and an independent Wishart (sample covariance) matrix.

1. First fix the parameters: $\mathrm{\Sigma }$ and d.f. $\nu$ (II).
2. Generate $\nu$ independent $N_p(0,\mathrm{\Sigma })$ vectors (III).
3. Form the sample covariance matrix $T$ (Wishart) from them (IV).
4. Finally construct $X^{\prime }{T}^{-1}X$, which follows the $T^2$ law (I).

Short trick: Parameters → data → covariance → statistic.

Concept: Kolmogorov: for independent zero-mean $x_i$ with finite variances, $P(\underset{1\le k\le n}{max}|S_k|\ge \epsilon )\le \frac{\text{Var}(S_n)}{{\epsilon }^2}$.

1. Start with the independent sequence (I).
2. Form the partial sums $S_k$ (III) – the inequality is about them.
3. Apply Kolmogorov's inequality (II).
4. Use it to bound the probability that some $S_k$ exceeds the threshold (IV).

Short trick: Variables → partial sums → inequality → conclusion about the maximum partial sum.

Concept: The basic principles of sample survey are statistical regularity (with inertia of large numbers), validity and optimisation.

1. Statistical regularity: a moderately large random sample reflects the population's characteristics.
2. Validity: the design must permit valid estimates and their errors (probability sampling).
3. Optimisation: maximum efficiency (minimum variance) for a given cost, or minimum cost for given precision.
4. 'Normality' is not a principle of sampling.

Short trick: R-V-O: Regularity, Validity, Optimisation.

Concept: Multicollinearity = near-linear dependence among the regressors (X's).

1. Montgomery–Peck list its sources: data collection method (sampling a narrow subspace), constraints on the model/population, model specification (e.g. polynomial terms), and an over-defined model.
2. Correlation between Y and the X's is desirable – it is what regression exploits, not a source of multicollinearity.
3. Hence a, c, d.

Short trick: Multicollinearity lives among X's only, never between Y and X.

Concept: Cyclical movements are isolated by removing trend and seasonal effects or by detecting periodicities.

1. Residual method: divide out T and S (and smooth I) to leave C.
2. Periodogram and harmonic analysis detect hidden periodicities / fit sine–cosine terms.
3. Method of monthly (simple) averages measures seasonal variation, not cycles.

Short trick: Monthly averages → seasonal; residual, periodogram, harmonic → cyclical.

Concept: Match each result with its content.

1. Borel σ-field: σ-field (closed under complements and countable unions) generated by open intervals (a–III).
2. Dominated convergence theorem: if $|f_n|\le g$, $g$ integrable, $f_n\to f$, then $lim\int f_n=\int lim{f}_n$ (b–I).
3. Kolmogorov's SLLN: ${\overline{X}}_n\to \mu$ almost surely (c–IV).
4. CLT: $\sqrt{n}({\overline{X}}_n-\mu )/\sigma \to N(0,1)$ – asymptotic normality (d–II).

Short trick: Strong LLN → a.s.; CLT → normal; DCT → limit inside integral.