INTRODUCTION TO PROBABILITY — DU B.Sc (hons) Statistics Previous Year Question Papers

7. (a) Anurn contains ncards marked from 1 ton. Two cards are drawn at a time. Find the mathematical expectation of the product of the numbers on the cards. (b) A random variable X has a distribution Find the distribution function and the pdf of 2. Attempt 5 questions in all. 4. Attempt 4 more questions selecting any two questions each from Section A and Section 8. (i) State classical and empirical definitions of probability and discuss their limitations. Explain how does the axiomatic approach overcome these limitations? (ii) For any three events A, 8 and C show that (iii) For n events A[, A2 ,..., An Show that p(U4)Sip(4) (iv) Three dice are thrown simultaneously. Find the probability of getting sum of the digits at least (v) A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart or a red card. (vi) A man is known to speak truth 3 out of 5 times. He throws a die and reports that the number obtained is a six. Find the probability that the number obtained is actually a six. (ii) Find the mean and variance of the above distribution. (71/2 x 2) 6. (a) What is the expectation of the numbers of failure preceding the first success in an infinite series 'of trials with constant probability p of success in each trial? (b) .The length of time (in minutes) that a certain lady speaks on telephone is found to be a random phenomenon, with probability function as specified below (i) Find the value of A that makes f(x) a pdf. (ii) What is the probability that number of minu.tes she will talk over the phone is (a) more than 5 minutes, (b) less than 10 minutes (c) between 5 and 10 minutes. (7]/2x2) 5. (a) The probability mass function of a random X is given by (iii) Distribution function F(`x) with largest x such that F(x) < 1/2. (b) The distribution of random variable X in the range (i) Verify the area under the curve is unity. (ix) A random variable X has the following probability mass function. Determine the value of k? Also evaluate P(X < 4), P(X 2 5) and P(3 < X S 6). What is the smallest value of x for which P(X S x) > I/2. In a lottery in tickets are drawn out of n tickets numbered from 1 to n. What is the expectation of the sum of the squares of numbers drawn? A filling station is supplied with gasolifle once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function 0 , otherwise what must the capacity of the tank be so that the probability of the supply's being exhausted in a given week is.01? (5x6) 2. (a) For independent events A£, i = 1, 2 ,..., n with respective probability of occurrence p;. Find the probability of occurrence of atleast one of them. (b) Distinguish between pair-wise and mutually independent events. If A, 8 and C are random events in a sample space and if A, 8 and C are pair-wise independent and A is independent of BUC, then show that A, 8 and C are mutually independent. (7]/2 x2) 3. (a) A, 8 and C al:e three urn which contain 2 white,