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Pune University - SY BSc (Sem - I) STATISTICS - II : Continuous Probab, April 2010

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Total No. of Questions : 4] P271 [Total No. of Pages : 4 [3717]-114 S.Y B.Sc. STATISTICS ST - 212 : Continuous Probability Distributions - I (Sem.-I) (Paper - II) (New Course) Time : 2 Hours] [Max. Marks : 40 Instructions to the candidates: 1) 2) 3) 4) All questions are compulsory. Figures to the right indicate full marks. Use of calculator and statistical tables is allowed. Symbols and abbreviations have thier usual meaning. Q1) Attempt each of the following : a) Choose the correct alternative in each of the following : i) If X is continuous random variable (r.v.) taking positive values then geometric mean (G.M.) is given by A) Antilog [E(X)] C) ii) [1 each] 1 Antilog E log X D) Antilog {E [x.log (X)]} B) Antilog {E [log (X)]} If X and Y are independent random variables then the conditional probability density function (p.d.f.) of Y given X = x is A) Marginal p.d.f. of X. B) Marginal p.d.f. of Y. C) Ratio of joint p.d.f. of X and Y to marginal p.d.f. of Y. D) Ratio of marginal p.d.f. of X to marginal p.d.f. of Y. iii) If Mx(t) = (1 2t) 8, t < A) G (2, 8) B) 1 then the probability distribution of X is 2 G (8, 2) C) 1 G 8, 2 D) 1 G , 8 2 P.T.O. b) State whether the given statement is true or false in each of the following: [1 each] i) For continuous r.v. X with p.d.f. f (x) and c.d.f. f (x), p (a < X < b) = f (b) f (a). ii) The points on the normal probability plot lying almost along straight line indicates that observations are from N ( , 2 ) distribution. iii) If X and Y are independent random variables then moment generating function (m.g.f.) of (2X + 6Y + 8) is e8t MX(2t) MY (6t). c) Let X be a continuous r.v. with c.d.f. F ( x) = d) 1 1 tan ( x ) + ; < x < find p.d.f. of X. 2 [1] Let X and Y be independent N (4, 4) and N (4, 5) variables respectively. State the mean of (X Y)2 . [1] 9 y 1+ x and E (Y | x) = then find the value of correlation 2 2 coefficient between X and Y. [1] e) If E (X | y) = f) State Lack of Memory Property of exponential distribution. Q2) Attempt any two of the following : [1] [5 each] a) If X P (m) then show that as m , probability distribution of X m tends to N (0, 1). m b) Let X be a continuous r.v. with p.d.f. f (x) = 6x (1 x) ; 0 < x < 1 = 0 ; otherwise find : i) harmonic mean ii) mode. c) The joint p.d.f. of continuous bivariate r.v. (X, Y) is f (x, y) = 12 xy (1 y) ; 0 < x < 1, 0<y<1 = 0 ; otherwise i) Find marginal probability distributions of X and Y. ii) Verify whether X and Y are independent. [3717]-114 2 Q3) Attempt any two of the following : a) i) ii) If X is continuous r.v. with p.d.f. f (x) and c.d.f. f (x) then obtain the probability distribution of Y = f (x). [3] The conditional p.d.f. Y given X is g 2 (Y | x ) = 2 ( x + y) ; 0 < y <1 2x + 1 = 0 ; otherwise find conditional mean of Y given X. [2] b) Find mean deviation about mean of Exp ( ) distribution. [5] c) The joint p.d.f. of bivariate random variable (X, Y) is f (x, y) = k.e (x + y) ; x > 0, y > 0 =0 ; otherwise. find : i) K, ii) P [X + Y > 1]. [5] Q4) Attempt any one of the following : a) i) If X 1 , X 2 , ..., X n a re independent identically distributed 2 N ( , ) variables, show that A) B) ii) Xi N (n 2 X N . n n i =1 ), [5] The p.d.f. of continuous r.v. X is f (x) = x 25 1 (10 x) 25 =0 find : A) E (X) = [3717]-114 2 ,n ; 0<x<5 ; 5 < x < 10 ; otherwise B) 3 P (3 < X < 8). [5] b) i) Find mean and variance of G ( ) distribution. [5] ii) If mean and variance of U (a, b) r.v. are 1 and 3 respectively then determine the values of a and b. [3] iii) Let X be a continuous r.v. with p.d.f. f (x) = 1 2 2 e 1 ( x 5 )2 8 = 0 find P [X > 5]. ; < x< ; otherwise [2] xxxx [3717]-114 4

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Additional Info : S.Y. B.Sc. Statistics (Semister - I), STT - 212 : Continuous Probability Distributions - I (Paper - II) (New Course), Pune University
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