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Pune University - FY BSc STATISTICS - II (Old Course)

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Total No. of Questions : 5] P233 [Total No. of Pages : 5 [3717] - 65 F.Y. B.Sc. STATISTICS / STATISTICAL TECHNIQUES Discrete Probability & Probability Distributions (Old Course) (Paper - II) Time : 3 Hours] Instructions to the candidates : 1) All questions are compulsory. 2) Use of statistical tables and calculator is allowed. 3) Figures to the right indicate full marks. 4) Symbols have their usual meanings. [Max. Marks : 80 Q1) Attempt the following: a) [8 x 2 = 16] Write down the sample space for the following random experiments. i) Counting of number of defective bulbs in a sample of 10 bulbs. ii) A student appears for an examination till he passes. b) What is probability of getting at least one head when two fair coins are tossed. c) Explain : Discrete random variable. d) Prove that variance of a constant is zero. e) State axioms of probability. f) Let A and B be two independent events, such that P(A) = find ( U ) . g) 1 1 , P(B) = , 4 3 Determine K, such that the following function is probability mass function (P.m.f.) of r.v.X. ( = x ) = Kx , x = 1, 2, 3, 4, 5 5 = 0, otherwise. h) Define Bernoulli distribution with parameter P, state mean of Bernoulli distribution. P.T.O. Q2) Attempt any four of the following: a) [4 x 4 = 16] Explain the following terms with one illustration each: i) Union of two events. ii) Intersection of two events. iii) Mutually exclusive events. b) If A and B are any two events defined on a sample space , then prove that P (A U B ) = P( A ) + P( B) P( A I B) . c) For the following probability distribution of r.v.x. X -2 -1 0 1 2 3 4 P(X = x) 0.10 0.10 0.10 0.20 0.30 0.15 0.05 Find : i) P (1 x 1 < 2) ii) P ( x 3 x > 0) iii) Mode of x. d) Define: i) ii) e) Pairwise independence of three events. Mutual independence of three events. Given ( ) = 1 5 2 , ( ) = and ( ) = 2 6 6 Find i) P(B) ii) iii) ( ) f) ( ) iv) ( ) . A random variable X has discrete uniform distribution with parameter n. Find mean and variance of X. Q3) Attempt any four of the following: a) [4 x 4 = 16] Let X be a discrete random variable with p.m.f. P(X = x) = x ; x = 1, 2, 3, 4, 5. 15 = 0, otherwise Find V(x). [3717] - 65 2 b) Let X H (N, M, n); find E(X). c) If X and Y are two discrete random variables, prove that E(X + Y) = E(X) + E(Y). d) Obtain recurrence relation for Binomial probabilities. e) For the following joint probability distribution: Y 1 1 1 2 2 2 3 4 X 3 Find 9 1 9 0 27 1 27 0 1 0 1 9 9 1 1 4 9 9 27 Marginal distribution of X and Y. ii) f) i) Conditional distribution on of X/Y = 3. For a bivariate discrete r.v. (x, Y): V(X) = 9, V(Y) = 4 and Cov (X, Y) = 4. Find i) V(2X + 3Y) ii) V(4X 3Y). Q4) Attempt any two of the following: [2 x 8 = 16] a) Define partition of a sample space. State and prove Baye s theorem. b) Define conditional probability P(A/B). Show that: ( ) () i) = 1 . ii) ( ) = ( ) i f A and B are mutually exclusive ( ) + ( ) events. [3717] - 65 3 c) Suppose A, B and C be three events defined on , such that 1 4 P(A) = P(B) = P(C) = ( ) = ( C ) = 0, ( C ) = 1 6 Calculate i) ( C ) . ii) ( ) iii) ( C ) iv) P ( C ) d) A fair coin is tossed three times. If X and Y denote the number of heads and number of runs respectively. i) Obtain joint probability distribution of (X, Y). ii) Obtain ( = 1) . Q5) Attempt any two of the following: a) [2 x 8 = 16] Define the following terms with one illustration each: i) Discrete sample space. ii) An event. iii) Complement of an event. iv) Impossible event. b) The joint probability distribution of (X, Y) is given below: X\Y 0 1 1 1 1 0 1 2 1 2 8 8 8 8 8 1 8 Find correlation coefficient between X and Y. [3717] - 65 4 c) The probability distribution of a discrete r.v.X is given below: X ii) d) 1 1 2 P(X) Find i) 2 1 1 1 1 3 6 6 3 first three raw moments. first three central moments also comment on the nature of skewness. i) State and prove additive property of Binomial distribution. ii) A box contains 4 red and 6 blue balls. 2 balls are chosen at random from the box without replacement. Find the probability that: [3] 1) No red ball is chosen. 2) Exactly 2 red balls are chosen. Y [3717] - 65 5 [5]

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Additional Info : F.Y.B.Sc. STATISTICS / STATISTICAL TECHNIQUES Discrete Probability & Probability Distributions (Old Course) (Paper - II), Pune University
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