BSc Statistics SEM I 2022 2023 Nov 2023 STATISTICS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Answer the following questions:
   • a) Correct the following if necessary: (10M)
   • (i) If A and B are mutually exclusive events, then AU B = @
   • (ii) Events A and B are independent iff P(A/B) = P(A)
   • (iv) The second raw moment about origin is variance
   • (v) Mean and standard deviation of discrete uniform distribution are equal
   • b) Answer in one sentence: (10M)
   • (i) If P(A U B) = P(B) = P(ANB) = find P(A)
   • (ii) Write down the classical of probability = (iii) If probability mass function of X is given below find P( X < 2.3)
   • (iv) If X and Y are independent random variables then what is
   • (v) State p.m.f of binomial distribution
2. Q2 Attempt any two from the following: (20M)
   • (i) State and prove the addition theorem on probability of two events. Also prove that if A and B are two events defined on sample space S, then
   • (ii) From a well shuffled pack of cards, 2 cards are selected at random, what ts the
   • a) both cards are red b) one is red and other is black c) both are aces
   • d) exactly one jack is drawn e) both are of the same suit
   • (iii) Define following events and give example for each
3. Q3 Attempt any two from the following: (20M)
   • (i) Define Random Variable, Discrete Random Variable, Probability Mass Function. Hence find i) P(X < 1) ii) P (X = —2) also obtain the probability distribution of Y = X? where probability mass function of a discrete random variable X ts given by
   • (ii) Let X be a discrete random variable with probability mass function p( x). a and b are constants. Then prove i) +b) +b
   • ii) V(X) = E - | Also derive expression for 3rd central
   • (iii)State and prove Multiplication Theorem of Expectation VCD/ FYBSC SEMI STATISTICS-II 3 100 marks Hence obtain the marginal probability mass function of X
   • ii) the marginal probability mass function of Y. For the joint p.m.f of random variables X and Y is given by
4. Q4 Attempt any two from the following: (20M)
   • (i) Let X be a random variable having Uniform Discrete Distribution assuming values {1, 2, ., Derive the expression for its mean and variance
   • (ii) (a) Derive the recurrence relation between probabilities for a random variable
   • (b) For a Binomial! Distribution mean is 3 and 15P(X = 0) = 2P(X 2 1)
   • (iii)(a)Write the p.m.f. of Hyper-geometric Distribution with parameters (N, M,n) and derive the expression for mean of Hyper-geometric Distribution
   • (b) If discrete random variable X has binomial distribution with parameters (6, p). Find p and q if 9P(X = 4) = P(X = 2)
5. Q5 Attempt any two from the following: (20M)
   • (i) A) A family has 2 children. Find the probability that both children are girls if it is known that:
   • (a) one of the children is girl
   • (b) the older child ts a girl
   • B) A ticket is drawn from the box containing 25 tickets and a number on it is observed. Obtain the probability that ticket drawn has a number
   • (a) less than 6 (b) greater than 20
   • (c) multiple of 5 (d) lying between 10 and 15, both inclusive
   • (ii) Define Bivariate Discrete Distribution, Marginal Probability Distribution, Conditional probability Distribution. X and Y are 2 random variables with joint probability distribution function as given below. Find a marginal probability mass function of X and Y also obtained conditional! probability distribution of X when Y = 6 and of Y when x = 2
   • (iii) A) If discrete random variable X has Uniform Distribution over the range {0,1,2,., n}. It's variance is 33.25. Find n and its mean
   • B) A taxi cab company has 10 Ambassadors and remaining 5 cars are of other A person wants to hire 7 taxis for a marriage party by random choice. Find the probability that he chooses
   • (a) Ail Ambassador (b) 3 Ambassadors than remaining

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