UPHESC Assistant Professor Statistics 2025 PYQs

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.

Concept: Identify the distinguishing feature of each design.

1. CRD uses only the two basic principles – randomisation and replication, no local control (a–II).
2. RBD groups units into homogeneous blocks so that variability between blocks is removed and comparisons are made within blocks (b–IV).
3. LSD controls two sources of variation – rows and columns (c–III).
4. BIBD ensures balance: every treatment is replicated equally (r times) and each pair occurs together equally often (d–I).
5. Only code (A) has a–II and c–III.

Short trick: CRD = R + R; RBD = blocks; LSD = rows + columns; BIBD = balance.

Concept: Standard multivariate tools and their roles.

1. Multivariate normal $N_p(\mu ,\mathrm{\Sigma })$ – joint distribution of several correlated variables (a–IV).
2. Hotelling's $T^2$ – tests a mean vector (generalised Student's t) (b–I).
3. Mahalanobis $D^2$ – distance between point/populations (c–II).
4. Wishart $W_p(n,\mathrm{\Sigma })$ – distribution of $\sum x_i{x}_i^{\prime }$, the matrix generalisation of ${\chi }^2$ (d–III).

Short trick: t → T², χ² → Wishart, distance → D².

Concept: When standards are not given, R-chart limits are based on the average range $\overline{R}$.

1. CL $=\overline{R}$, UCL $=D_4\overline{R}$, LCL $=D_3\overline{R}$.
2. $D_3=1-3d_3/d_2$ (0 for $n\le 6$), $D_4=1+3d_3/d_2$.
3. Single $R$ is not the centre line; LCL with $D_4$ would equal UCL.

Short trick: R chart: $D_3\overline{R}$ – $\overline{R}$ – $D_4\overline{R}$.

Concept: Under $H_0$, $\frac{n-p}{p(n-1)}{T}^2\sim F_{p,n-p}$, so the correct rejection rule is $T^2>\frac{p(n-1)}{n-p}F_{p,\,n-p,\,\alpha}$(needs$n>p$).

1. Rejection is in the upper tail, so (A) and (C) with '<' are wrong.
2. (D) $T^2>F_{p,n-p,\alpha}$ drops the whole scaling factor – wrong.
3. (B) $T^2>\frac{n-1}{n-p}F_{p,n-p,\alpha}$hastherightdirectionbutomitsthefactor$p$;itiscorrectonlywhen$p=1$, not in general.
4. So, as printed, no option gives the correct general rule; the question is treated as a bonus. The right criterion is $T^2>\frac{p(n-1)}{n-p}F_{p,\,n-p,\,\alpha}$.

Short trick: $T^2=\frac{p(n-1)}{n-p}F$ – remember the 'p' in front; for $p=1$, $T^2=t^2=F_{1,n-1}$.

Concept: Consistency: $T_n\stackrel{P}{\to }\theta$ as $n\to \mathrm{\infty }$ – defined only in the limit of large samples.

1. Unbiasedness and sufficiency are fixed-$n$ (small-sample) properties.
2. Efficiency is usually compared for given $n$ (asymptotic efficiency is a separate notion).
3. Hence consistency is the large-sample property.

Short trick: Consistency = 'gets closer to θ as n grows'.

Concept: By WLLN, ${\overline{X}}_n\stackrel{P}{\to }\mu =4$.

1. Chebyshev: $P(|\bar X_n-4|>0.006)\le\frac{1}{n(0.006)^2}\to0$.
2. $\{\bar X_n>4.006\}\subset\{|\bar X_n-4|>0.006\}$, so its probability → 0.
3. 0.5 would be the answer only for the threshold exactly 4 (by CLT).

Short trick: Any fixed gap above μ → probability 0; exactly at μ → ½.

Concept: $\sum c_i{T}_i$ is a contrast iff $\sum c_i=0$.

1. a: $3+1-3+1=2\ne 0$ – not a contrast.
2. b: $-3-1+1+3=0$ ✓
3. c: $1+2-2-1=0$ ✓
4. d: $1-1+1-1=0$ ✓

Short trick: Add the coefficients – zero means contrast.

Concept: Key life-table identities: $e_x^0=\frac{T_x}{l_x}$, $q_x=\frac{d_x}{l_x}=\frac{2m_x}{2+m_x}$, $T_x=L_x+T_{x+1}$.

1. Complete expectation of life $e_x^0=T_x/l_x$ ⇒ a–IV. Only code (D) contains a–IV.
2. $T_x$ is the cumulative person-years lived beyond age $x$, built up recursively from $T_{x+1}$ ⇒ d–II.
3. The mortality-rate column is $d_x/l_x$ (c–III), and its expression through the central death rate is $\frac{2m_x}{2+m_x}$ (b–I).
4. So the matching is a–IV, b–I, c–III, d–II.

Short trick: $e_x^0=T_x/l_x$ – spot this first and the code follows.

Concept: With $2^r$ blocks there are $2^r-1$ degrees of freedom between blocks, so $2^r-1$ effects are confounded in all.

1. $r$ of them are chosen independently.
2. The rest are their generalised interactions, confounded automatically: $2^r-1-r$.
3. E.g. $r=2$ (4 blocks): ABC and CDE chosen ⇒ ABDE confounded automatically: $2^2-2-1=1$ ✓.

Short trick: Total confounded $2^r-1$; minus the r chosen = $2^r-r-1$.

Concept: A moving average smooths the series: it irons out short-term fluctuations and leaves the long-term trend.

1. Seasonal and other short-period oscillations (and irregular movements, largely) are averaged out, especially when the period of the moving average equals the period of the oscillation.
2. The long-term variation is the trend itself – it is what the method measures, not what it removes, so 'all the above' is wrong.
3. Cyclical swings are removed only if their period coincides with the M.A. period – a special case, not the general effect.

Short trick: Moving average = smoothing filter: short-term out, trend in.