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Pune University - SY BSc (Sem - I) STATISTICS - I, April 2010

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Total No. of Questions : 4] P270 [Total No. of Pages : 3 [3717] - 113 S.Y. B.Sc. STATISTICS ST - 211 : Discrete Probability Distributions and Time Series (Sem. - I) (Paper - I) (New Course) Time : 2 Hours] [Max. Marks : 40 Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. 3) Use of calculator and statistical tables is allowed. 4) Symbols and abbreviations have their usual meaning. Q1) Attempt each of the following: a) Choose the correct alternative in each of the following: i) [1 Each] if the probability mass function (p.m.f.) of a discrete random variable (r.v.) is 1 x p(x) = k , 4 =0 , x = 1, 2, 3, . . . . . . . otherwise then the value of k is A) B) 4 3 C) ii) 3 4 3 D) 1 3 If X is a Poisson r.v. with variance 2 then fourth central moment of X is A) 2 B) 6 C) 12 D) 14 iii) If X NB (k, p) such that E(X) = 16 and Var (X) = 80 then A) k = 4, p = 1 5 B) k = 4, p = 4 5 C) 1 3 D) k = 8, p = 2 3 k = 8, p = P.T.O. b) State whether the given statement is true or false in each of the following: [1 Each] i) Median of discrete probability distribution may not be unique. ii) If X and Y are independent and identically distributed (i.i.d.) geometric random variables then X + Y is a geometric random variable. iii) Irregular variations are predictable in the analysis of time series. c) State range set of r.v.X denoting number of specified interval of time. - particles emitted in the [1] d) Define (r, s)th raw moment of bivariate discrete r.v. (X, Y) where r and s are non - negative integers. [1] e) If MX(t) = e3.2 (et 1) is moment generating function (m.g.f.) of r.v.X then state the modal value of X. [1] f) Explain the term calling population used in queuing theory. Q2) Attempt any two of the following: a) [1] [5 Each] Define the following terms: i) A discrete r.v.X defined on countably infinite sample space. ii) Probability distribution of a discrete r.v.X. iii) Mathematical expectation of a discrete r.v.X. b) The joint p.m.f. of bivariate discrete r.v. (X, Y) is p(x, y) = q2 py 2 , x = 1, 2, . . . . . . , y 1 y = 2, 3, 4, . . . . . . 0 < p < 1, q = 1 p =0 , otherwise. Find (i) conditional distribution of X given Y = y and (ii) E(X | Y = y). c) Let X and Y be independent Poisson random variables with means 2 and 4 respectively. Find (i) P(X + Y = 4) and (ii) P(Y = 3 | X + Y = 6). [3717] - 113 2 Q3) Attempt any two of the following: a) [5 Each] Let X be a discrete r.v. with p.m.f. p(x) = qx p , x = 0, 1, 2, . . . . . . . 0 < p < 1, q = 1 p =0 , otherwise. Find mean and variance of X. b) If X P(m) then show that, for a positive integer K, mk P(X > K) < . k! c) A departmental store operates with a single cashier. Customers arrive at a rate of 20 per hour. Average number of customers processed by the cashier is 24 per hour. Assuming Poisson process, find i) the probability that cashier is idle, ii) the average number of customers in the queuing system, iii) the average time a customer spends in the system. Q4) Attempt any one of the following: a) i) Describe the method of ratio of moving averages for computing seasonal indices. Discuss its merits and demerits. [6] ii) With usual notation, show that A) Max + b (t) = ebt MX (at), B) b) MX+ Y (t) = MX(t) MY (t) where X and Y are independent random variables. [4] i) Explain the fitting of autoregression models AR (1) and AR (2) to the time series data. [5] ii) A scientist inoculates mice, one at a time, with a disease germ until he finds 2 mice that have contracted the disease. If the probability 1 of contracting the disease is , what is the probability that at the 6 most 4 mice are required to be inoculated? [5] Y [3717] - 113 3

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Additional Info : S.Y. B.Sc. Statistics (Semister - I), ST - 211 : Discrete Probability Distributions and Time Series (Paper - I) (New Course), Pune University
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