UPSC Statistics 2023

(a) Out of 1000 persons born, only 900 reach the age of 15 years, and out of every 1000 who reach the age of 15 years, 950 reach the age of 50 years. Out of every 1000 who reach t…

(a) Let X, Y, Z be three mutually independent standard exponential variates and W₁ = X + Y + Z, W₂ = (X + Y)/(X + Y + Z), W₃ = X/(X + Y). Then (i) determine the joint distribution…

(a) (i) If X is a random variable with finite variance, show that lim (n→∞) n² P{|X| > n} = 0. (ii) In a certain recruitment test, there are multiple choice questions. There are f…

(a) What is the role of properties of completeness and sufficiency in Statistical Inference ? Explain. In U (0, θ), find out Uniformly Minimum Variance Unbiased Estimator (UMVUE)…

(a) (i) If **X** = (X₁ X₂ X₃)' is distributed as N₃ (μ, Σ), find the distribution of [(X₁ – X₂) (X₂ – X₃)]'. (5 marks) (ii) Suppose that **X** = (X₁ X₂ X₃)' ~ N₃ (**0**, Σ), where…

(a) Let **X** = (X₁ X₂ X₃)' be distributed as N₃ (μ, Σ), where μ = (2 −1 3)' and Σ = [[4, 1, 0], [1, 2, 1], [0, 1, 3]]. Find (i) the conditional distribution of (X₁ X₂)' given X₃…

(a) Discuss the difference between sampling for variables and sampling for attributes with examples. For a qualitative characteristic, find an unbiased estimator of population pro…

(a) Differentiate between randomised block design and balanced incomplete block design. In usual notations, for a balanced incomplete block design, prove that (i) bk = vr (ii) λ(v…

(a) What do you understand by Statistical Quality Control (SQC)? Discuss briefly its need and utility in Industry. Discuss the causes of variation in quality. (10 marks) (b) Consi…

(a) What do you understand by control chart for fraction defective? Explain its construction. Give the theoretical distribution on which the control limits are based. (15 marks) (…

(a) Solve the following Linear Programming problem using Two Phase method : Maximize Z = 3x₁ - x₂ Subject to 2x₁ + x₂ ≥ 2, x₁ + 3x₂ ≤ 2, x₂ ≤ 4, x₁ ≥ 0, x₂ ≥ 0 (b)(i) Solve the fo…

(a) A Company ships truckloads of grain from three silos to four mills. The supply (in truckloads) and the demand (also in truckloads) together with the unit transportation costs…

(a) State and discuss the important points which have to be considered in the construction of index numbers. (10 marks) (b) In a study of its sales, a motor company in Delhi obtai…

(a) Explain the principle of least squares. How it is used in fitting trend in time series analysis ? Explain the fitting of trend for the curve y=ab^tc^t². 15 marks (b) Define st…

7.(a) Derive, by starting from a suitable functional form for lₓ, the formula (i) Lₓ = (lₓ + lₓ₊₁)/2 and (ii) Lₓ = (lₓ - lₓ₊₁)/((log lₓ - log lₓ₊₁)) = -(dₓ)/(log pₓ) (iii) eₓ⁰ = 1…

8.(a) Explain the method of collection of agriculture data. Describe the (i) official publications for data collection and (ii) statistics collected by the various official agenci…