Answer precise answers.

46. QUESTION: An airfreight company has various classes of freight. In one of these classes the average weight of packages is 10kg and the variance of the weight distribution is 9kg-. Assuming that the package weights are independent (it is not the case that a single company is sending a large number of identical packages, for instance), estimate the probability that 100 packages will have total weight more than 1020kg.24. QUESTION: Tay-Sachs disease is a rare fatal genetic disease occurring chiefly in children, especially of Jewish or Slavic extraction. Suppose that we limit ourselves to families which have (a) exactly three children, and (b) which have both parents carrying the Tay-Sachs disease. For such parents, each child has independent probability - of getting the disease. Write X to be the random variable representing the number of children that will have the disease. (a) Show (without using any knowledge you might have about the binomial distribution!) that the probability distribution for X is as follows: 9 P(X = k) (b) Show that E( X) = = and that Var(X ) = 1625. QUESTION: A six-sided die has four green and two red faces and is balanced so that each face is equally likely to come up. The die will be rolled several times. Suppose that we score 4 if the die is rolled and comes up green, and 1 if it comes up red. Define the random variable X to be this score. Write down the distribution of probability for X and calculate the expectation and variance for X.45. QUESTION: Suppose that of 1,000,000 live births in Paris over some period, 508,000 are boys. Suppose X is Bin(106, 0.5) and calculate approximately P[X 2 508, 000). Does it seem reasonable to you that the proportion of males among Parisian babies conceived soon after the above period will be 50%. (Laplace developed his limit theorem in the late 1700's to deal with a question similar to this.)30. QUESTION: A machine produces items of which 1% at random are defective. How many items can be packed in a box while keeping the chance of one or more defectives in the box to be no more than 0.5? What are the expected value and standard deviation of the number of defectives in a box of that size?34. QUESTION: Experiments by Rutherford and Geiger in 1910 showed that the munber of alpha particles emitted per unit time in a radioactive process is a random variable having a Poisson distribution. Let X denote the count over one second and suppose it has mean 5. What is the probability of observing fewer than two particles during any given second? What is the P(X 2 10)? Let Y denote the count over a separate period of 1.5 seconds. What is P(Y > 10)? What is P(X + Y 2 10)