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

1. (a) The following table represents the running times (in minutes) of 26 randomly selected recent American movies. Construct a stem-and-leaf plot and comment on interesting features of the display. (b) The following data represents the actual blood pressure values for nine randomly selected individuals. Compute the median, upper fourth, and lower fourth. (c) Following data consists of value of sodium content in one serving of cereal for a sample of cereals manufactured by certain company. (i) Calculate sample mean, sample variance, and sample standard deviation. (ii) If 50 is added to each observation of the sample to obtain the sample of transformed observations. Compute the sample variance and sample standard deviation of transformed observations without reperforming the calculations. 2. (a) The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is 0.4, and the analogous probability for the second signal is 0.5, and the probability that he must stop at at least one of the two signals is 0.7. What is the probability that he must stop (i) At both signals? (il) At the first signal but not at the second one? (ili) At exactly one signal? (b) State Baye's theorem. The reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1-2 pages), medium (3-4 pages), or long (5-6 pages). Data on recent reviews indicates that 60% of them are short, 30% are medium, and the other 10% are long. Reviews are submitted in either Word or LaTex. For short reviews, 80% are in word, whereas 50% of medium reviews are in Word and 30% of long reviews are in Word. Suppose a recent review is randomly selected. (i) What is the probability that the selected review was submitted in Word format? (ii) If the selected review was submitted in Word format, what is the posterior probability of it being short? (c) An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A) = 0.4, and P(B) = 0.7. (i) What is the probability that at least one of the two projects will be successful? (ii) Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful? 3. (a) A certain brand of upright freezer is available in three different rated capacities: 450 L, 500 L and 550 L. Let X is the rated capacity of a freezer of this brand sold at a certain store. Suppose that has probability mass function (pmf) p(x) with p(450) = 0.2, p(500) = 0.5 and p(550) = 0.3. If the price of a freezer having capacity X is 2.5X - 650, what is the expected price and variance price paid by the next customer to buy a freezer?