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Pune University - FY BSc STATISTICS - II, April 2010

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Total No. of Questions : 5] [Total No. of Pages : 5 [3717] - 14 P192 F.Y. B.Sc. STATISTICS / STATISTICAL TECHNIQUES Discrete Probability and Probability Distributions (Paper - II) (New Course) Time : 3 Hours] Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. 3) Use of statistical tables and calculator is allowed. 4) Symbols have their usual meanings. Q1) a) [Max. Marks : 80 Choose correct alternative for the following : [4 x 1 = 4] i) If A and B are independent events with P(A) = 0.5 and P(B) = 0.2, then P ( A B) is ii) I) 0.7 II) 0.6 III) 1.0 IV) 0.52 If random variable X has binomial distribution with parameters n and p, then I) mean > variance II) mean = variance III) mean < variance IV) mean variance. iii) For a Poisson distribution if P[ X = 2 ] = 1 P[ X = 3] , then the 2 parameter of the distribution is equal to I) 2 II) 3 III) 6 IV) 5 iv) For a sample space = {w1 , w2 , w3 , w4 } , P{w1} = P{w2} = 2 , 8 1 , P{w4} = k. 8 This will be a probability model if k is equal to I) 0 II) 1 P{w3} = III) 1 8 IV) 3 8 P.T.O. b) c) d) e) f) State whether the following statements are true or false : [4 x 1 = 4] i) For two discrete random variables X and Y, E(X + Y) = E(X) + E(Y), only if X and Y are independent. ii) If B A , then P(A/B) = 1. iii) Mode of a random variable X is the maximum value of X. iv) Mean and variance of discrete uniform distribution are equal. Define discrete sample space. [2] State any two properties of distribution function. [2] Define joint probability mass function of two dimensional discrete [2] random variable. Explain with illustration what is meant by a Bernoulli trial. [2] Q2) Attempt any four of the following : [4 x 4 = 16] a) Explain the following, with one illustration each : i) Mutually exclusive events. ii) Exhaustive events. b) Given the following distribution function of a random variable X : X 1 2 3 4 5 F(x) 0.10 0.35 0.65 0.80 1.00 Find i) Probability mass function of X. ii) Median of X. iii) Mode of X. c) Suppose that the number of cars X that pass through a washing centre between 5.00 p.m. to 6.00 p.m. on a particular day has the following distribution : X 4 5 6 7 8 9 1 2 2 3 2 2 12 12 12 12 12 12 Let 9(X) = 12X, represent the amount of money in rupees, paid to the attendent by the manager. Find the attendent s expected earning for the time period specified above. Give the classical definition of probability. State its limitations. If A and B are any two events defined on , then prove that, P ( A B) P(A) + P(B) . Also state Boole s inequality for k events A1, A2, ........ Ak. P[X = x] d) e) [3717] -14 2 f) Define : i) Conditional probability. ii) Pairwise independence of three events. Q3) Attempt any four of the following : [4 x 4 = 16] a) State and prove the multiplication theorem for two events A and B defined on a sample space . Also state its generalisation for three events A, B and C. b) If X and Y are independent random variables with the following distributions : X 1 0.6 P[X = x] 2 Y 0.4 5 15 0.2 P[Y=y] 10 0.5 0.3 Find the joint distribution of X and Y. c) If P(A) = 3 1 11 , P(B) = and P ( A B) = , 4 3 12 Find i) ii) d) P(A B) P(exactly one of A and B occurs). Let X be a discrete r.v. with following probability distribution : X 0 1 2 3 P[X = x] 2 8 1 8 4 8 1 8 Find E[X2]. e) A random variable X has the following probability distribution : X 1 P[X = x] 0 2 3 0.15 0.20 0.35 0.30 i) ii) f) [3717] -14 Find P[ X 0 ]. Find probability distribution of Y = 2X 1. Show that all raw moments of a Bernoulli (p) r.v. are equal to p. 3 Q4) Attempt any two of the following : a) i) State and prove the additive property of binomial random variables. [5] ii) Consider following sample space of English alphabets = {a, b, c, d, ........ y, z} List the elements of the following events. A = { x x isa vowel } B = { x x precedes e in alphabet} Also, answer the following : Are A and B mutually exclusive? b) i) Define : 1) ii) Mutual independence of three events. 2) c) [3] Partition of a sample space. [4] State the p.m.f. of a H(N, M, n) random variable and obtain its [4] mean. For the following joint probability distribution of (X, Y), compute the correlation coefficient between X and Y [ (X, Y)]. Y 1 2 3 1 1 8 0 2 8 2 2 8 1 8 2 8 X d) [8] Define a discrete uniform probability distribution. Give two real life situations where it can be applied. [4] ii) [3717] -14 i) 1 For a Bernoulli r.v. X, 3 = 0.6 . Find its mean, variance and third central moment. [4] 4 Q5) Attempt any two of the following : a) [2 x 8 = 16] Let X be a discrete r.v. with mean and variance 2 , then prove that, i) [2] ii) Var(aX) = a2 2 [3] iii) Var(aX + b) = a2 2 . b) Var(X + b) = 2 [3] The following table gives the joint probability distribution of X and Y : Y 0 1 2 1 2 12 1 12 3 12 2 1 12 2 12 0 3 1 12 1 12 1 12 X Find i) E(X 2Y) ii) Var(Y/X = 2) [8] c) Prove that under certain conditions to be stated, binomial distribution tends to Poisson distribution. [8] d) Define : i) A discrete random variable. ii) The (r, s)th raw moment of a bivariate distribution. iii) rth order factorial moment of a discrete r.v. iv) Probability generating function (p.g.f) of a discrete r.v. [3717] -14 5 [8]

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