Probability for Computing — DU B.Sc. (Hons) Computer Science Previous Year Question Papers

3. Parts of the questions to be attempted together. (a) Prove that Var (2i=1 Xi) = 2L, Var (Xi) 4 (b) Given the cumulative distribution function: 4 (c) Suppose that X is a random variable with mean 10 and variance 15. 4 i) Calculate the bound on probability P(5 ≤X ≤ 15) using Chebyshev's inequality? ii) What can we say about P{ X ≥ 20} using Markov's inequality? (d) What is meant by the periodicity of a class in Markov chain. Identify the 4 periodicity of states in the following Markov chain: (e) Suppose that a trainee soldier shoots a target in an independent fashion. If the 4 probability that the target is hit in any shot is 0.7. What is the probability that it takes him fewer than four shots to hit the target? (D) The joint probability mass function of X and Y, p(x,y), is given by: Compute E[X|Y = i] for i = 1,2, 3. Assume the experiment of throwing two dice and answer the following: 5 (g) i. Give the number of outcomes in the sample space for the experiment. il. Give the favorable outcomes for the event that the number on the first die is greater than 5. iii. Find the probability of getting a sum of the numbers on two dice greater than or equal to 10, given that the number on the first die is 4 2. Derive the expression for the expected value of a geometric distribution. 5 (b) Let (Xn}nzo be a discrete-time Markov chain on the state space {1, 2, 3} with 5 one-step transition probability matrix: And the initial distribution: Then calculate the value of P(X, = 2, X2 = 3, X3 = 1) (rounded off to three decimal places. (c) Differentiate between mutually independent and pairwise independent events. 5 Let the equally likely outcomes of an experiment be one of the four points in the three-dimensional space with Cartesian coordinates (1, 0, 0), (0, 1, 0), (0, 0, 1), and (1, 1, 1). Let us define the events A, B and C as: A: X-coordinate value is one. B: Y-coordinate value is one. C: Z-coordinate value is one. Verify if these events are mutually-independent or not? (a) A random sample of size 100 is taken from a population whose mean is 60 & 7 variance is 400. Using central limit theorem, with what probability can we assert that the mean of the sample will not differ from mean by more than 4? Given that (b) For a continuous random variable, write the expression for condition probability density function of Random variable Y given that X is defined. Let f(x,y)=x+yfor0≤x 51andOSy 51.Find fx/x (y/x). 4. Give the probability mass function for the Poisson distribution. Let X & Y be 7 (a) independent Poisson random variables with respective means 1, & 1z. Calculate the distribution of X+Y. (b) Consider a Markov chain with state space S = (1,2,3,4,5) The transition probability matrix is given by: 0 0 0.6 0.5-i. Construct the corresponding state transition diagram for the above Markov chain. ii. Identify the communicating classes in the diagram ili. Classify each state as: Transient or Recurrent iv. Determine whether the chain is Ergodic or not?