UPSC Statistics 2025

(a) Let E, F and G be three pairwise independent events such that P(E∩F) = 0·1 and P(F∩G) = 0·3. Prove that P(Eᶜ∪G) ≥ 11/12. 10 marks (b) If X and Y are non-negative independent r…

(a) Let X be a continuous random variable having probability density function f(x) = (2/25)(x+2), -2 ≤ x ≤ 3; 0, otherwise. Find the cumulative distribution function of Y = X² and…

(a) Let probability of obtaining Head on a biased coin be 4/5 and X be the number of heads obtained in a sequence of 25 independent tosses of the coin. The same coin is tossed aga…

(a) Let X₁, X₂, ... be a sequence of random variables from Bernoulli distribution with mean θ, 0<θ<1. Derive SPRT for testing H₀ : θ = θ₀ versus H₁ : θ = θ₁ = 1 – θ₀, 0<θ₀<1. Also…

(a) For a two variable linear regression model Yᵢ = a + bXᵢ + eᵢ, where E(eᵢ) = 0, Var(eᵢ) = σ²ₑ, Cov(eᵢ, eⱼ) = 0 for i ≠ j, (i,j) ∈ {1, 2, ..., n}, if â and b̂ are least square e…

(a)(i) If (X, Y) follows bivariate normal BN(μ₁, μ₂, σ₁², σ₂², ρ), then obtain (A) E(e^X) (B) E(e^(X+Y)) (C) Var(e^X) and (D) Correlation between e^X and e^Y. 3+3+3+3=12 marks (a)…

(a) Analyse and interpret the following data concerning output of wheat per field obtained as a result of experiment conducted to test four varieties of wheat A, B, C and D under…

(a)(i) What are principal components ? Show that the principal components are uncorrelated. (10 marks) (a)(ii) Obtain the principal components and the amount of variation explaine…

(a) State the significance of operating characteristic (OC) curves in control chart analysis. Obtain the general expression for the OC function corresponding to the mean (X̄) char…

(a) (i) What are control charts by variables and control charts by attributes? 5 marks (ii) Derive the control limits for the construction of control charts for the mean and varia…

(a) A company manufactures 30 items per day. The sale of those items depends upon demand which has the following distribution : | Sale (units) | 27 | 28 | 29 | 30 | 31 | 32 | |---…

(a) Solve the game whose payoff matrix is -1 & -2 & 8 7 & 5 & -1 6 & 0 & 12 (15 marks) (b) Use the penalty (Big M) method to solve the following linear programming problem : Minim…

(a) Explain the multicollinearity problem in a regression model. What are its consequences? State the different indicators of multicollinearity and explain. 10 marks (b) Establish…

(a) On the basis of the figures given below, calculate the age-specific death rates (ASDRs) for all the age groups. Also, calculate the crude death rate (CDR) on the basis of ASDR…

(a) Explain the concept of index number. Calculate the Fisher's ideal index number from the following data and verify that whether it satisfies time reversal and factor reversal t…

(a) Define time series. For a moving-average process with weights {a₁, a₂, ..., aₘ} of random components {eᵢ, i = 1, 2, ...}, where eᵢ's are i.i.d. N(0, σ²), obtain the correlogra…