BSc Statistics SEM I 2019 2020 Nov 2020 STATISTICS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 A) State whether the following statements are true or false .If true justify the Statement .If false correct it. (10)
   • 1) If P(A/B) = P(A) then A depends on B and are two random variables with V(X) =2 and =3
   • 4) For a discrete random Variables XE(X) (x?)
   • 5) The binomial distribution js symmetric when p>q in One sentence (10)
   • 1) Define Binomial distribution
   • 2) State multiplication theorem
   • 3)If cumulative Distribution functions of discrete random variable X is Give Statistical definition of Probability
   • 5) If A and B are independent events then show that P(AUB) = P(A) + P(A)P(B) Attempt any two sub questions (10 marks each question) (20)
   • A) State and Prove that BAYES Theorem,
   • B) A man has 5 One rupees coins and one of them is known to have two heads. He Select one coin at and tosses it 5 times. If it always falls head upwards, what is the Probability that it is the coin with two heads?
   • 2) State and Prove multiplication theorem on Probability hence find P(A/B) and P(AUB) if P(A) == and P(B) =
   • 3) i) Give any mathematical definition of a Probability ang State its limitations li) Define the following
   • a) Deterministic experiments
   • b) Objective
   • c) Complimentary Event
   • 4)A committee of four is to be formed among 3 teachers of Arts faculty 4 Of Science faculty ,2 of Commerce faculty and 1 from self-financing courses .Find the probability that the 4 faculties are represented in the committee committee will have the teacher from self financing course and at least teacher from Arts faculty, The committee will have no teacher from self-financing and exactly | teacher from
2. Q3 Attempt any two sub questions (10 marks each question) Define mathematical expectation Discrete random variables X 20 marks
   • ii) Show that minimum when a= (x) the effect of change of origin and scale of expectation
   • 2. i) Define between X and Y. b) Correlation coefficient between X and Y Further state what are they if X,Y are independent ? suitable illustration explain the following for di screte random variable X and Y
   • a) Joint protability mass function of X and Y
   • b) Marginal probability mass function of X and Y
   • c) Conditional probability mass function of Y given X =x
   • 3.i) State and prove that the additional theorem on expectation for two random variable
   • ii) Show that COV(X,Y) =E (XY) - E(y)
   • iii) X and Y are two stochastically independence random variables with means 10 &12 and 9&16 respectively compute (a) E(7x-6y+4) (b)v(4x-3y+2) (c)
   • 4. A fair coin is tossed 4 times let X denotes the number of heads Occurring in the last two tosses and Y denote the total number of heads occurring in the four tosses. F ind the probability distribution of X and Y. Also obtain the conditional probability distribution of Y given X= lalso find E(xy)
3. Q4 Attempt any two sub questions (10 marks each question) 20 marks
   • 1. Define hyper geometric distribution derive its mean and variance
   • 2. i) Describe important features of distribution, Also find recurrence relation for the probabilities of binominal distribution li)An unbiased coin is tossed five times. What is the probability of getting at least: two
   • 3.Define Poisson Distribution derived its variance also important features of Poisson
   • 4.Define discrete uniform distribution hence solve random variable X has probability mas
4. Q5 Attempt any four of the following (Five marks each question) 20 marks
   • 1. Derive Mean and variance of Bernoulli Distribution
   • 2. For Poisson approximation to Binominal distribution prove that
   • 3. A discrete variable X has the following probability distribution Obtained the first four row moments about origin. Calculate coefficient of skewness and kurtosis and comment
   • 4. A Random variable X takes values 1,3,5 with probabilities and = Respectively. Cumulative function, F (x) of X hence obtain F (1-3),F(6) also draw the graph of cumulative distribution function and Bare independent events then prove that i) A and B ii) A and B and iii) A and B. A and Bare independent
   • 6. Two fair dice are rolled simultaneously. Write the sample space of the experiment Find the probability that the sum of the number of the upper most faces is (i)8, (ii) less than S,
   • (iii) multiple of 4 mass function of a discrete random variable X is given by Find P(x>-1) Obtain the probability distribution of y = x?

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