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State whether each of the following random variables is discrete or continuous: a. The number of defective tires on a car b. The body temperature of a hospital patient c. The number of pages in a book d. The number of draws (with replacement) from a deck of cards until a heart is selected e. The lifetime of a light bulb

Classify each of the following random variables as either discrete or continuous: a. The fuel efficiency (miles per gallon) of an automobile b. The amount of rainfall at a particular location during the next year c. The distance that a person throws a baseball d. The number of questions asked during a 1-hour lecture e. The tension (in pounds per square inch) at which a tennis racket is strung f. The amount of water used by a household during a given month g. The number of traffic citations issued by the highway patrol in a particular county on a given day

6.9 Two six-sided dice, one red and one white, will be rolled. List the possible values for each of the following random variables. a. \(x=\) sum of the two numbers showing b. \(y=\) difference between the number on the red die and the number on the white die (red - white) c. \(w=\) largest number showing

Consider the random variable \(y=\) the number of broken eggs in a randomly selected carton of one dozen eggs. Suppose the probability distribution of \(y\) is as follows: \(\begin{array}{cccccc}y & 0 & 1 & 2 & 3 & 4 \\ p(y) & 0.65 & 0.20 & 0.10 & 0.04 & ?\end{array}\) a. Only \(y\) values of \(0,1,2,3,\) and 4 have probabilities greater than \(0 .\) What is \(p(4) ?\) b. How would you interpret \(p(1)=0.20 ?\) c. Calculate \(P(y \leq 2)\), the probability that the carton contains at most two broken eggs, and interpret this probability. d. Calculate \(P(y<2),\) the probability that the carton contains fewer than two broken eggs. Why is this smaller than the probability in Part (c)? e. What is the probability that the carton contains exactly 10 unbroken eggs? f. What is the probability that at least 10 eggs are unbroken?

Suppose that fund-raisers at a university call recent graduates to request donations for campus outreach programs. They report the following information for last year's graduates: $$\begin{array}{lllll}\text { Size of donation } & \$ 0 & \$ 10 & \$ 25 & \$\end{array}$$ 0.30 0 Proportion of calls 0.45 .20 0.05 Three attempts were made to contact each graduate. A donation of $$\$ 0$$ was recorded both for those who were contacted but declined to make a donation and for those who were not reached in three attempts. Consider the variable \(x=\) amount of donation for a person selected at random from the population of last year's graduates of this university. a. Write a few sentences describing what donation amounts you would expect to see if the value of \(x\) was observed for each of 1000 graduates. b. What is the most common value of \(x\) in this population? c. What is \(P(x \geq 25)\) ? d. What is \(P(x>0)\) ?

Suppose that \(20 \%\) of all homeowners in an earthquakeprone area of California are insured against earthquake damage. Four homeowners are selected at random. Define the random variable \(x\) as the number among the four who have earthquake insurance. a. Find the probability distribution of \(x\). (Hint: Let \(S\) denote a homeowner who has insurance and \(\mathrm{F}\) one who does not. Then one possible outcome is SFSS, with probability (0.2)(0.8)(0.2)(0.2) and associated \(x\) value of 3 . There are 15 other outcomes.) b. What is the most likely value of \(x ?\) c. What is the probability that at least two of the four selected homeowners have earthquake insurance?

A pizza shop sells pizzas in four different sizes. The 1000 most recent orders for a single pizza resulted in the following proportions for the various sizes: $$ \begin{array}{lcccc} \text { Size } & 12 \text { in. } & 14 \text { in. } & 16 \text { in. } & 18 \text { in. } \\ \text { Proportion } & 0.20 & 0.25 & 0.50 & 0.05 \end{array} $$ With \(x=\) the size of a pizza in a single-pizza order, the given table is an approximation to the population distribution of \(x\). a. Write a few sentences describing what you would expect to see for pizza sizes over a long sequence of single-pizza orders. b. What is the approximate value of \(P(x<16)\) ? c. What is the approximate value of \(P(x \leq 16)\) ?

A contractor is required by a county planning department to submit anywhere from one to five forms (depending on the nature of the project) when applying for a building permit. Let \(y\) be the number of forms required of the next applicant. Suppose the probability that \(y\) forms are required is known to be proportional to \(y ;\) that is, \(p(y)=\) \(k y\) for \(y=1, \ldots, 5\) a. What is the value of \(k ?\) (Hint: \(\Sigma p(y)=1 .)\) b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required?

6.27 The article "Probabilistic Risk Assessment of Infrastructure Networks Subjected to Hurricanes" (12th International Conference on Applications of Statistics and Probability in Civil Engineering, 2015) suggests a uniform distribution as a model for the actual landfall position of the eye of a hurricane. Consider the random variable \(x=\) distance of actual landfall from predicted landfall. Suppose that a uniform distribution on the interval that ranges from \(0 \mathrm{~km}\) to \(400 \mathrm{~km}\) is a reasonable model for \(x\). a. Draw the density curve for \(x\). b. What is the height of the density curve? c. What is the probability that \(x\) is at most \(100 ?\) d. What is the probability that \(x\) is between 200 and \(300 ?\) Between 50 and \(150 ?\) Why are these two probabilities equal?

A particular professor never dismisses class early. Let \(x\) denote the amount of additional time (in minutes) that elapses before the professor dismisses class. Suppose that \(x\) has a uniform distribution on the interval from 0 to 10 minutes. The density curve is shown in the following figure: a. What is the probability that at most 5 minutes elapse before dismissal? b. What is the probability that between 3 and 5 minutes elapse before dismissal?