Mathematical Statistics — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Type of the Paper : DSE 1. (a) A large but sparsely populated country has two small hospitals, one at the south end of the country and the other at the north end. The south hospital's emergency room has 4 beds, whereas the north hospital's emergency room has only 3 beds. Let X denote the number of south beds occupied at a particular time on a given day, and let Y denote the number of north beds occupied at the same time on the same day. Suppose that these two rvs are independent, that the pmf of X puts probability masses. 1, 2, 3, 2 and. 2 on the x values 0, 1, 2, 3, and 4, respectively, and that pif of Y distributes probabilities .1, 3, 4, and .2 on the y values 0, 1, 2, and 3, respectively. (1) Display the joint pmf of X and Y in a joint probability table. (i) Compute P(X s 1 and Y s 1) and verify that this equals the product of (i) Express the event that the total number of beds occupied at the two hospitals combined is at most 1 in terms of X and Y, and then calculate this probability. (iv) What is the probability that at least one of the two hospitals has no beds occupied ? (b) An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. Compute the covariance for X and Y. (ii) Compute p for X and Y. (c) A concert has three pieces of music to be played before intermission. The time taken to play each piece has a normal distribution. Assume that the three times are independent of each other. The mean times are 15, 30, and 20 min, respectively, and the standard deviations are 1, 2, and 1.5 min, respectively: What is the probability that this part of the concert takes at most one hour? Are there reasons to question the independence assumption? Explain. (a) A system consisting of two components will continue to operate only as long as both components function. Suppose the joint pof of the lifetimes (months) of the two components in a system is given by Ax, y) = c [1O - (x + y)], for x > O, y > 0, * + y < 10. Find the value of the constant c. If the first component functions for exactly 3 months, what is the probability that the second functions for more than 2 months. (b) Define the bivariate normal distribution. For a few years, the SAT consisted of three components : writing, critical reading, and mathematics. Let W = SAT Writing score and X = SAT Critical Reading score for a randomly selected student. According to the College Board, in 2012 W had mean 488 and standard deviation 114, while X had mean 496 and standard deviation 114. Suppose X and W have a bivariate normal distribution with Corr(X, W) = 5. If an English department plans to use X + W., a student's total score on the nonmath sections of the SAT to help determine admission. Then determine the distribution of X + W.