Probability and statistics.
1. A survey of people in given region showed that 25% drank regularly. The probability of death due to liver disease, given that a person drank regularly, was 6 times the probability of death due to liver disease, given that a person did not drink regularly. The probability of death due to liver disease in the region is 0.005. If a person dies due to liver disease what is the probability that he she drank regularly? 2. In a production line ICs are packed in vials of 5 and sent for inspection. The probabilities that the number of defectives in a vial is 0,1,2,3 are 1/3, 1/4, 1/4, 1/6 respectively. Two ICs are drawn at random from a vial and found to be good. What is the probability that all ICs in this vial are good? 3. An insurance company floats an insurance policy for an eventuality taking place with probability 0.05 over the period of policy. If the sum insured is Rs. 100000 then what should be the premium so that the expected earning of the insurance company is Rs. 1000 per policy sold? 4. Suppose X is a discrete random variable with puf given by P(X =-1) = _-", P(X=0) =2, P(X=1) =", where a is a real number. Find the range of a for which this is a valid pmf. Also determine the maximum and minimum values of Var (X). Write the of for a = . Hence show that the median is not unique. 5. Let X be a continuous random variable with the pdf bx f(x) = (1+x)' Ve-1
2| X =0.25) P(X 1). 18. From a store containing 2 defective, 3 partially defective and 3 good computers, a random sample of 4 computers is selected. Find the expected number of defective and partially defective computers in the sample. What will be the covariance between the number of defective and number of partially defective computers? 19. Let (X, Y) have bivariate normal distribution with density function f(x y) = - 30 0 . Define random variables Y1, Y2 and Y3 as Y1 =X1 + X2 + X3, Y2 = X1 + X2 X1 X1 + X2+X3' "X1 +X2 Find the joint and marginal densities of Y1, Yz and Ya. Are they independent? 21. Suppose that for a certain individual, calorie intake at breakfast is random variable with mean 500 and standard deviation (s.d.) 50, calorie intake at lunch is a random variable with mean 900 and s.d. 100, and calorie intake at dinner is a random variable with mean 2000 and s.d. 180. Assuming that intakes at different meals are independent of one another, what is the approximate probability that the average calorie intake per day over the next year (365 days ) is at most 3410? 22. Let X,, Xy, X, Xar be independent N(0, 1) random variables and Y= -X X? . Find the distribution of X,H / VY. 23. Let X,, Xy.., Xz, be independent N(0,1) random variables and 24(24-X2) . What is the distribution of W?1. In a game, a contestant is shown two identical envelopes containing money. The contestant does not know how much money is in either envelope. The contestant can choose an envelope at random (equally likely), see how much money is in it, and then either keep this amount or exchange the chosen envelope for the other one, keeping the money in the second envelope. Now suppose that one envelope has $1000, and the other has $2000. A strategy to play this game is for the contestant to fix a number r, and if the first envelope is revealed to have
