4: 1. Classify each random variable as either discrete or continuous: (a) The time between customers entering a checkout lane at a retail. (b) The weight of refuse on a truck arriving at a landfill. (c) The number of passengers in a passenger vehicle on a highway at rush hour. (d) The number of clerical errors on a medical chart. (e) The number of accident-free days in one month at a factory. (f) The air pressure of a tire on an automobile. (g) The amount of rain recorded at an airport one day . (h) The number of students who actually register for classes at a university next semester. 2. A discrete random variable X has the following probability distribution: X 13 18 20 24 27 P(x) 0.22 0.25 0.2 0.17 0.16 Compute each of the following quantities (give the exact numbers): (a) P(18)= (b) P(X>18)= (c) P(X18)= (d) The mean of X is (e) The variance 2 of X is (write all decimal points) (f) The standard deviation of X is (use 4 decimal points) 3. Borachio works in an automotive tire factory. The number X of sound but blemished tires that he produces on a random day has the probability distribution X 2 3 4 5 P(x) 0.48 0.36 0.12 0.04 For the following, don't round the numbers. (a) Find the probability that Borachio will produce more than three blemished tires tomorrow. (b) Find the probability that Borachio will produce at most two blemished tires tomorrow. (c) Compute the mean of X. 4. An insurance company will sell a $10,000 one-year term life insurance policy to an individual in a particular risk group for a premium of $368. Find the expected value to the company of a single policy if a person in this risk group has a 97.25% chance of surviving one year. 5. Find the mean and standard deviation of a random variable X with the following probability distribution 1 X 0 P(X) 0.3 (a) =1.5, =0.96 (b) =1.2, =0.96 (c) =1.5, =0.98 (d) =1.2, =0.98 (e) =1.5, =2.4 1 0.3 2 0.3 3 0.1 6. To set a schedule for a new bus route, the transit authority repeatedly times the trip between two points; the time X in minutes is found to have the following probability distribution function X 20 21 22 23 24 25 P(x) 0.07 0.19 0.36 0.24 0.1 0.04 (a) the probability that a randomly selected trip will take at least 24 minutes is about: (i) 0.14; (ii) 0.10; (iii) 0.23; (iv) 0.04; (v) 0.86 (b) The probability that a randomly selected trip will take between 21 and 24 minutes (including 21 and 24) is about: (i) 0.19; (ii) 0.79; (iii) 0.89; (iv) 0.29; (v) 0.7 (c) If the trip is made over and over, the average time it takes in minutes is about: (i) 22.5; (ii) 22.2; (iii) 21.9; (iv) 22.0; (v) 23.0 7. A sociologist surveyed the households in a small town. The random variable X represents the number of dependent children in the households. The missing probability is X 0 1 2 3 4 P(X) 0.07 0.2 0.38 ? 0.13 (a) 0.02 (b) 0.12 (c) 0.22 (d) 0.32 (e) 0.78 8. The following is a probability distribution function. True or False? X 0 1 2 3 4 5 6 P(X) 0.3 0.25 0.25 0.1 0.05 0.03 0.02 X 0 1 2 3 4 5 P(X) 3/4 1/10 1/20 1/25 1/50 -1/100 X 0 1 2 3 P(X) 0.005 0.435 0.555 0.206 2 X P(X) X P(X) 0 0.05 1 0.25 2 0.35 3 1.25 4 -0.9 0 0.17 1 0.1 2 0.35 3 0.2 4 0.16 9. Determine whether or not the random variable X is a binomial random variable. (a) X is the number of dots on the top face of a fair die. (b) X is the number of hearts in a five-card hand drawn (without replacement) from a well-shuffled ordinary deck. (c) X is the number of defective parts in a sample of ten randomly selected parts coming from a manufacturing process in which 0.02% of all parts are defective. (d) X is the number of times the number of dots on the top face of a fair die is even in six rolls of the die. (e) X is the number of dice that show an even number of dots on the top face when six dice are rolled at once. (f) X is the number of black marbles in a sample of 5 marbles drawn randomly and with replacement from a box that contains 25 white marbles and 15 black marbles. 10. An English-speaking tourist visits a country in which 30% of the population speaks English. He needs to ask someone directions. The tourist sees four local people standing at a bus stop. Find the probability that at least one of them will be able to speak English (use the Binomial distribution and use 4 decimal points for the answer) 11. Adverse growing conditions have caused 5% of grapefruit grown in a certain region to be of inferior quality. Grapefruit are sold by the dozen. (a) Find the average number of inferior quality grapefruit per box of a dozen. (b) A box that contains two or more grapefruit of inferior quality will cause a strong adverse customer reaction. Find the probability that a box of one dozen grapefruit will contain two or more grapefruit of inferior quality (use 4 decimal points). 3