UPSC Statistics 2022

(a) Let X and Y be independent random variables with exponential distribution having respective means 1/(λ₁) and 1/(λ₂), λ₁ > 0, λ₂ > 0. Find E [max (X, Y)]. (10 marks) (b) Using…

(a) Let a random variable X have exponential distribution with mean 1/θ, θ > 0. To test H₀ : θ = 3 against H₁ : θ = 2, construct sequential probability ratio test. Show that proba…

(a) (i) How large a sample must be taken in order that the probability will be at least 0·90 that the sample mean will be within 0·4 – neighbourhood of the population mean, provid…

(a) Consider Poisson distribution P_θ(X = j) = (e⁻θ θ^j)/(j!) = pⱼ, j = 0, 1, 2, .... Let fⱼ be the frequency for X = j and E(fⱼ) = mⱼ = npⱼ. Discuss how you obtain minimum chi-sq…

(a) Define general linear model with usual assumptions. If y₁ = β₁ + u₁, y₂ = –β₁ + β₂ + u₂, y₃ = –β₂ + u₃, where u₁, u₂, u₃ are mutually independent random variables with mean ze…

(a) In a set of two-way classified data according to k levels of factor A and r levels of factor B, there is one observation in each cell. Show that the total number of error cont…

(a) Consider the following data given for a BIBD with v = b = 4, r = k = 3, λ = 2 and N = 12 : Analyse the design. [Given that : F₃,₅ (0·05) = 5·41] 15 (b) (i) The data matrix of…

(a) (i) What are orthogonal polynomials ? How do you fit an orthogonal polynomial of degree 'p' ? (ii) For the model Y_(n×1) = X_(n×k) β_(k×1) + u_(n×1), E(uu') = σ² I_n, where X_…

(a) Distinguish between process control and product control. Explain the various sources of variation encountered in a process control study. Suggest how they can be eliminated fr…

(a) Samples of size n = 5 units are taken from a process every hour. The x̄ and R̄ values for a particular quality characteristic are determined. After 25 samples have been collec…

(a) It is planned to raise a research team to a strength of 50 chemists, which is to be maintained. The wastage of recruits depends on their length of service which is as follows:…

(a) Use penalty method to solve the following linear programming problem : Maximize Z = x₁ + 2x₂ + 3x₃ - x₄ subject to the constraints x₁ + 2x₂ + 3x₃ = 15 2x₁ + x₂ + 5x₃ = 20 x₁ +…

(a) Apply the method of link relatives to the following data and calculate the seasonal indices : Price of Rice (in ₹ per 10 kg) | Quarter | 2001 | 2002 | 2003 | 2004 | |---------…

(a) Explain Akaike information criterion for order selection in an ARMA (p, q) process. 15 marks (b) Define autocorrelation coefficient. What are its consequences for ordinary lea…

(a) What are the various indices of mortality measure? Explain the purpose and procedure for standardizing them. (20 marks) (b) With usual notations, obtain logistic curve as give…

(a) How does the concept of wholesale price index work? Describe the major components of wholesale price index. Explain the methodology of index numbers of area, production and yi…