BSc Sem VI 2022 2023 Apr 2023 STATISTICS DISTRIBUTION THEORY STOCHASTIC PROCESS Question Paper - Mumbai University | munotes

Questions asked in this paper

• Figures to the right indicate full marks

1. Q1 OF derive marginal distribution of X and marginal distribution of
   • (b) Suppose (X, Y) follows Bivariate Normal Distribution with parameters (0 p). Obtain its joint MGF and hence marginal distributions of X and 10
   • (p) Define fisher’s Z transformation and explain its use in testing 10
2. Q1 Ho: p = po against Hi: p > po li. Ho: p = po against Hi: p < po Ho: p = po against Hi: p # po for large samples, in case of bivariate Also obtain 95% confidence interval for p
   • (q) Let X, Y and Z Normal(0,1). 10
3. Q1 Show that Joint distribution of U = X + aZ and V = Y + aZ is Bivariate il. In above example, find ‘a’ such that correlation coefficient between U
4. Q2 X be r.v. such that = n=0,1,2. Obtain generating functions of the following sequences 10 marks
   • (b) i. Define Probability Generating Function (PGF) of discrete random variable Obtain PGF of Negative Binomial distribution and hence derive its mean and (08) 2
   • (p) i.A uniform die is thrown. Let X denote number on uppermost face of adie. Find 4
   • ii. Bernoulli trial is repeated till success is occurred K times. Let X be the number of tra=ilas required to get success. Obtain PGF of X and mean 6
   • (q) x, ben independent random variables with PGFs ,P2(S), Pn(S) respectively. Obtain the PGF of x; ul. Let U(S) and V(S) be the generating functions of two independent random variables, respectively. Obtain the generating functions of (a) X + Y and (b) X — 10
5. Q3 (a) In the usual notations, for Poisson birth process, list the postulates, state the expression P, (t), the probability of ‘n’ numbers in the system at time ‘t’ and find its mean and variance 10 marks
   • (b) Stating clearly postulates for the pure death process, with initially ‘1’ members in the system at time t=0. Derive the difference differential equation for Ln = nu and find the expression for the probability of ‘n’ numbers in the system at time ‘t’ 10
   • (p) For linear growth model having birth rate (nA) and death rate (nw) and initially at time t = 0, there are ‘a’ member in the system then 10
6. Q1 List the Postulates
   • (q) Inusual notation for Poisson death process, list the postulates, state the expression for P,(t), probability that ‘n’ members in the system at time ‘t’. Find its mean 10
   • (a) Define the following terms; 10
   • (i) Input Process
   • (ii) Server and customer
   • (iii) Reneging (iv)
   • (v) Balking
   • (b) Show that for a single service station with Poisson arrival and exponential service time the probability that exactly ‘n’ calling units are in the queue system is, Where, p is the traffic intensity 10
   • (p) For : (N/FIFO)} queueing model, derive the expression for P, and find 10
   • (q) Discuss the classification of queuing model, and operating characteristics of 10
7. Q5 Attempt Any Two sub questions
   • (a) Show that (X, Y) follows Bivariate Normal Distribution with parameters (Uy of if and only if every linear combination of X and Y viz. ax+by, , b+0, is a Normal Variate. Where a and b are constants 10
   • (b) Let X be ar.v. assuming values 0, 1, 2, . with probabilities po, p2,. Let qj = P(X and pj = P(X =j),j 1, 2, . Assuming actual notations, show that Q(S) = where Q(S) = and P(S) = Also obtain mean and variance of X in terms of Q 10
   • (c) For pure birth process, if the growth rate is directly proportional to the of individuals (n > 1) say 2, n, assuming no death or removal and initially at time t=0, there are ‘1’ members in the system, then 10
   • ii. Find P, (t)
   • (d) What are the service disciplines? Describe some forms of common service Disciplines and illustrate with examples 10

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