UPHESC Assistant Professor Statistics 2025 PYQs

Concept: Match each wetland to its actual state.

1. Sakhya Sagar (Shivpuri) – Madhya Pradesh (2022) ✔.
2. Sirpur Wetland (Indore) – Madhya Pradesh, not Tripura ✘.
3. Pala Wetland (Saiha) – Mizoram, not Meghalaya ✘.
4. Sur Sarovar / Keetham Lake (Agra) – Uttar Pradesh (2020) ✔ → 1 and 4.

Short trick: Indore has two Ramsar sites – Sirpur and Yashwant Sagar; Mizoram's only one is Pala.

Concept: The Aihole inscription (634 CE) of Chalukya king Pulakeshin II was composed by the Jain poet Ravikirti.

1. In it Ravikirti claims to have attained the fame of Kalidasa and Bharavi – the earliest dated mention of both poets.
2. Prayag Prashasti (Harishena) praises Samudragupta; Bhitari pillar is of Skandagupta; Ayodhya inscription is of Dhanadeva (Shunga).

Short trick: Aihole → Ravikirti → "Kalidasa-Bharavi"; also records Pulakeshin II defeating Harsha.

Concept: During the Zulu (Bambatha) Rebellion of 1906 in Natal, Gandhi raised an Indian Stretcher-Bearer Corps.

1. For the duration of service he was given the temporary rank of Sergeant Major (his autobiography, Part IV).
2. He and his men mostly nursed wounded Zulus, an experience that pushed him towards brahmacharya and Satyagraha.
3. In the Boer War (1899) he had led the Indian Ambulance Corps.

Short trick: Boer War 1899 – Ambulance Corps; Zulu Revolt 1906 – Sergeant Major.

Concept: Babur wrote his memoirs, Tuzuk-i-Baburi (Baburnama), in his mother tongue Chagatai Turkish.

1. It was translated into Persian by Abdur Rahim Khan-i-Khanan in 1589 at Akbar's order.
2. English translations: Leyden & Erskine (1826), A. S. Beveridge (1922).

Short trick: Babur – Turki original; Rahim – Persian translation.

Concept: Public Interest Litigation originated in the USA in the 1960s (civil-rights and legal-aid movements), where it was used to represent unrepresented interests.

1. In India it was developed in the late 1970s–80s by Justices P. N. Bhagwati and V. R. Krishna Iyer by relaxing locus standi.
2. Landmark Indian cases: Hussainara Khatoon (1979), S. P. Gupta (1981).

Short trick: PIL – born in the USA, popularised in India by Bhagwati & Krishna Iyer.

Concept: The Constituent Assembly met for the first time on 9 December 1946 in the Constitution Hall (Central Hall), New Delhi.

1. Dr. Sachchidananda Sinha was the temporary (oldest) President; Dr. Rajendra Prasad was elected President on 11 December 1946.
2. Objectives Resolution moved by Nehru on 13 December 1946; Constitution adopted on 26 November 1949.

Short trick: 9-12-1946 first sitting → 26-11-1949 adoption → 26-1-1950 commencement.

Concept: Article 326: elections to the Lok Sabha and State Legislative Assemblies shall be on the basis of adult suffrage.

1. Voting age reduced from 21 to 18 years by the 61st Amendment Act, 1988.
2. Art. 324 – Election Commission; Art. 325 – single general electoral roll, no exclusion on religion, race, caste or sex; Art. 327 – Parliament's power to make election laws.

Short trick: 324 Commission, 325 one roll, 326 adult vote, 327 Parliament's law.

Concept: As per FAO world production statistics available when the paper was set, China was the largest producer of rice (paddy) and India the second.

1. India is, however, the largest exporter of rice (about 40% of world trade).
2. Note for later readers: USDA estimates for 2024-25 placed India's milled-rice output (~150 Mt) above China's (~145 Mt), so later sources call India No. 1.

Short trick: Rice: China 1st producer (FAO), India 2nd producer but No. 1 exporter.

Concept: The Production Linked Incentive (PLI) scheme (2020–21) gives incentives on incremental sales of goods made in India in 14 key sectors.

1. Sectors include large-scale electronics/mobile manufacturing, automobiles & auto components, medical devices, pharmaceuticals, telecom, food processing, textiles, solar modules, ACC batteries, specialty steel, drones, white goods, IT hardware.
2. All three given sectors are covered, so "None of the above".

Short trick: PLI started with mobiles, medical devices and pharma (2020), then expanded to 14 sectors.

Concept: Post-1947 land reforms aimed at social justice and higher productivity by giving land to the tiller.

1. Measures: abolition of zamindari/intermediaries, tenancy reforms, land ceiling and redistribution of surplus land to the landless, consolidation of holdings.
2. Real estate, agro-industries or privatisation were never the primary aim.

Short trick: "Land to the tiller" – abolish zamindari, fix ceiling, redistribute surplus.

Concept: Events with no common sample point satisfy $A\cap B=\phi$; such events are called mutually exclusive (disjoint).

1. No common point ⇒ $A\cap B=\phi$ ⇒ $P(A\cap B)=0$.
2. Complementary needs also $A\cup B=S$ – not given.
3. Independence means $P(A\cap B)=P(A)P(B)$; disjoint events with positive probabilities are in fact dependent, but the defining name is 'mutually exclusive'.

Short trick: No overlap = mutually exclusive; overlap with product rule = independent.

Concept: $A\cap B\subseteq A$ and $A\cap B\subseteq B$, so $P(A\cap B)\le min\{P(A),P(B)\}$.

1. Joint occurrence is a sub-event of each event, hence its probability cannot exceed either individual probability.
2. E.g. independent events: $P(A\cap B)=P(A)P(B)$, a product of two fractions – smaller than each.
3. It is 1 or 0 only in special cases, so 'smaller' is the general answer.

Short trick: Both together is harder than either alone – probability gets smaller.

Concept: For $B(n,p)$: mean $np$, variance $npq$.

1. $q=\frac{npq}{np}=\frac{4}{8}=\frac{1}{2}$, so $p=\frac{1}{2}$, $n=16$.
2. $P(X=1)=(\genfrac{}{}{0ex}{}{16}{1}){\left(\frac{1}{2}\right)}^{16}=16\cdot 2^{-16}=2^4\cdot 2^{-16}$.
3. $={\left(\frac{1}{2}\right)}^{12}$.

Short trick: q = variance/mean; then $np{q}^{n-1}$.

Concept: A variable that can take any value in an interval is continuous; one taking countable values is discrete.

1. Height is measured (e.g. 165.37 cm) and can take any value in a range.
2. Counts like number of children are discrete – height is not a count.
3. Hence height is a continuous random variable.

Short trick: Measured → continuous; counted → discrete.

Concept: $nb(x;r,p)=(\genfrac{}{}{0ex}{}{x+r-1}{r-1})p^r{q}^x$ = number of failures before the $r$-th success.

1. Put $r=1$: $(\genfrac{}{}{0ex}{}{x}{0})p{q}^x=p{q}^x$, $x=0,1,2,\dots$
2. This is the geometric distribution (failures before the first success).
3. Poisson is the limit as $r\to \mathrm{\infty }$ with $rq\to \lambda$, not $r=1$.

Short trick: Waiting for 1st success = geometric; for r-th success = negative binomial.

Concept: $M(t_1,t_2)=E(e^{t_{1}X+t_{2}Y})=\sum _{r,s}{\mu }_{rs}^{\prime }\frac{t_1^r{t}_2^s}{r!s!}$.

1. Expand $e^{t_{1}X}{e}^{t_{2}Y}=\sum _r\frac{(t_{1}X{)}^r}{r!}\sum _s\frac{(t_{2}Y{)}^s}{s!}$.
2. Taking expectation gives $E(X^r{Y}^s)$ multiplied by $\frac{t_1^r{t}_2^s}{r!s!}$.
3. So ${\mu }_{rs}^{\prime }$ = coefficient of $\frac{t_1^r{t}_2^s}{r!s!}$ (true for any bivariate MGF, including normal).

Short trick: Univariate: coefficient of $t^r/r!$; bivariate: of $t_1^r{t}_2^s/(r!s!)$.

Concept: S.D. is independent of origin but not of scale: if $u=x/h$, then ${\sigma }_u={\sigma }_x/|h|$.

1. Here $h=10$, so new S.D. $=\sigma /10$.
2. Variance would become ${\sigma }^2/100$ – option (B) confuses S.D. with variance.

Short trick: Divide data by k → S.D. ÷ k, variance ÷ k².

Concept: Karl Pearson: $S_k=\frac{\text{Mean}-\text{Mode}}{\sigma }$.

1. $-0.4=\frac{60-\text{Mode}}{45}$ ⇒ $60-\text{Mode}=-18$.
2. Mode $=60+18=78$.

Short trick: Negative skew ⇒ mode > mean: Mode = Mean − $S_k\sigma$.

Concept: ${\beta }_{yx}=\frac{\text{Cov}(X,Y)}{{\sigma }_x^2}$.

1. Independence ⇒ $E(XY)=E(X)E(Y)$ ⇒ Cov$(X,Y)=0$.
2. Hence ${\beta }_{yx}=0$ (regression line of Y on X is horizontal).

Short trick: Independent ⇒ r = 0 ⇒ both regression coefficients = 0.

Concept: $L(p)$ = probability of accepting a lot of quality $p$.

1. If $p=0$, the sample can contain no defective, so the lot is surely accepted.
2. Hence $L(0)=1$ (and $L(1)=0$); a probability can never be $\pm \mathrm{\infty }$.

Short trick: Perfect lot → always accepted → L(0) = 1.