RMI 3750 - Problem Set 2, Due 10/11/2017 1. Two jars contain coins. Jar I contains 7 pennies, 6 nickels and 8 dimes. Jar II contains 5 pennies, 3 nickels and 1 dime. A jar is selected at random and a coin is selected from that jar. If the coin is a nickel, what is the probability that it came from jar II? 2. A card is drawn from a standard 52 cards deck, not replaced, and a second card is drawn. What is the probability that the second card is a spade? 3. A probability distribution for the claim sizes for an auto insurance policy is given in the table below Claim Size Probability 20 0.20 30 0.10 40 0.10 50 0.20 60 0.15 70 0.10 80 0.15 a) Evaluate the cdf for this distribution at the following points: 0, 20, 30, 40, 50, 90. 1 b) What is the mean of this distribution? c) What is the variance of this distribution? d) What percentage of the claims are within one standard deviation of the mean claim size? (In other words, what percentage of the claims are greater than or equal to [the mean size minus one standard deviation] but less than or equal to [the mean size plus one standard deviation]?) 4. Define a random variable Y as the outcome of a fair die roll times 3 (e.g., if a five is rolled, Y takes a value of 5*3=15). What is the expected value of Y? What is the variance of Y? 2 5. A random variable X has E(X)=3 and V(X)=7. Define a new random variable: Y=6X + 3. What is E(Y) and V(Y)? 6. In class we often talked about a 6-sided die. Now consider a 24-sided die, with sides numbered 1 to 24 (so that the sample space of outcomes is {1,2,. . . ,24}). Suppose that all outcomes are equally likely. What is the mean and variance of the outcome of a roll of a 24-sided die? 7. A firm's revenue distribution for the coming year is (in million dollars): 15% probability of 5, 25% probability of 10, 20% probability of 12, 30% probability of 14 and 10% probability of 20. If F(.) is the cumulative distribution function of this random variable, what is F(11) equal to? What is F(14)? What is the variance of the firm's revenue distribution? 3 8. Airtran's flight #307 can accomodate 50 passengers, but the flight is overbooked, as 52 tickets were sold. Each ticketed passenger can arrive late and miss the flight with a probability 0.02. What is the probability that no passenger arrives late? What is the probability that exactly one passanger arrives late? What is the probability that Airtran has to pay overbooking fees (and reschedule passengers to different flights)? 9. The world incidence of (frequency of world population having) diabetes is 5%. If 20 persons are chosen at random, what is the probability that no more than 3 have the disease? 10. Plot in Excel the probability mass function (density) for the following: a Binomial random variable X with n = 10, p = 0.3 a Binomial random variable Y with n = 100, p = 0.4 a Uniform random variable X with n = 50 Attach the clearly labeled graphs. 4 11. Suppose we know that births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability that we observe 5 births in a given 2 hour interval? 12. In hospital A births occur randomly at an average rate of 2.3 births per hour and in hospital B births occur randomly at an average rate of 3.1 births per hour. What is the probability that we observe 7 births in total from the two hospitals in a given 1 hour period? 13. A pediatrician wishes to recruit 5 couples, each of whom is expecting their 5 first child, to participate in a new natural childbirth regimen. Let p be the probability that a randomly selected couple agrees to participate. If p = .2, what is the probability that 15 couples must be asked before 5 are found who agree to participate? 14. An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. What is the probability that the first strike comes on the third well drilled? 6 15. Boxes contain 20 items of which 10% are defective. Find the probability that no more than 2 defectives will be obtained in a sample of size 10. Contrast your results to what you would obtain if you use a binomial distribution (sampling with replacement), rather than the (correct) hypergeometric. 7