After measuring the duration of many telephone calls, the telephone company found their

In Problems 14, given that p(x) is a density function, find the value of a.

In Problems 14, given that p(x) is a density function, find the value of a.

In Problems 14, given that p(x) is a density function, find the value of a.

In Problems 14, given that p(x) is a density function, find the value of a.

In Problems 56, graph a density function representing the given distribution. The age at which a person dies in a society with high infant mortality and in which adults usually die between age 40 and age 60.

In Problems 56, graph a density function representing the given distribution. The heights of the people in an elementary school.

Match the graphs of the density functions (a), (b), and (c) with the graphs of the cumulative distribution functions I, II, and III. (a) (I) (b) (II) (c) (III)

In Problems 810, graph a density function and a cumulative distribution function which could represent the distribution of income through a population with the given characteristics A largemiddle class.

In Problems 810, graph a density function and a cumulative distribution function which could represent the distribution of income through a population with the given characteristics Small middle and upper classes and many poor people.

Small middle class, many poor and many rich people.

An insect has a life-span of no more than one year. Figure 7.28 shows the density function, p(t), for the lifespan. (a) Do more insects die in the first month of their life or the twelfth month of their life? (b) What fraction of the insects live no more than 6 months? (c) What fraction of the insects live more than 9 months? 12 1 6 t (months) fraction of population per month p(t) Figure 7.28

A large number of people take a standardized test, receiving scores described by the density function p graphed in Figure 7.29. Does the density function imply that most people receive a score near 50? Explain why or why not. 10 20 30 40 50 60 70 x test scores fraction of students per test score Figure 7.29

Figure 7.30 shows the distribution of the number of years of education completed by adults in a population. What does the shape of the graph tell you? Estimate the percentage of adults who have completed less than 10 years of education. 8 12 16 t (years of education) p(t) % of population per year of education Figure 7.30

Figure 7.31 shows P(t), the percentage of inventory of an item that has sold by time t, where t is in days and day 1 is January 1. (a) When did the first item sell? The last item? (b) On May 1 (day 121), what percentage of the inventory had been sold? (c) Approximately what percentage of the inventory sold during May and June (days 121181)? (d) What percentage of the inventory remained after half of the year had passed (at day 181)? (e) Estimate when items went on sale and sold quickly. 60 120 180 240 300 360 50 100 t (days) % of inventory sold P(t) Figure 7.31

The density function for radii r (mm) of spherical raindrops during a storm is constant over the range 0 < r < 5 and zero elsewhere. (a) Find the density function f(r) for the radii. (b) Find the cumulative distribution function F(r).

Major absorption differences have been reported for different commercial versions of the same drug. One study compared three commercial versions of timed-release theophylline capsules.9 A theophylline solution was included for comparison. Figure 7.32 shows the cumulative distribution functions P(t), which represent the fraction of the drug absorbed by time t. Which curve represents the solution? Compare absorption rates of the four versions of the drug.

In southern Switzerland, most rain falls in the spring and fall; summers and winters are relatively dry. Sketch possible graphs for the density function and the cumulative distribution function of the rain distribution over the course of one year. Put the date on the horizontal axis and fraction of the years rainfall on the vertical axis.

After measuring the duration of many telephone calls, the telephone company found their data was well approximated by the density function p(x) = 0.4e0.4x, where x is the duration of a call, in minutes. (a) What percentage of calls last between 1 and 2 minutes? (b) What percentage of calls last 1 minute or less? (c) What percentage of calls last 3 minutes or more? (d) Find the cumulative distribution function.

Find the median of the density function given by p(t) = 0.04 0.0008t for 0 t 50 using the Fundamental Theorem of Calculus.

In 1950 an experiment was done observing the time gaps between successive cars on the Arroyo Seco Freeway. The data10 show that, if x is time in seconds and 0 x 40, the density function of these time gaps is p(x) = 0.122e0.122x. Find the median and mean time gap. Interpret them in terms of cars on the freeway.

If p(10) = 1/2, then half the population has x < 10.

If P(10) = 1/2, then half the population has x < 10.

If p(10) = 1/2, then the fraction of the population lying between x = 9.98 and x = 10.04 is about 0.03.

If p(10) = p(20), then none of the population has x values lying between 10 and 20.

If P(10) = P(20), then none of the population has x values lying between 10 and 20.

If a quantity has density function p(x), then the mean value and median are never equal.

The normal distribution with density function p(x) = (1/2)ex2/2 has mean = 0 and standard deviation = 1.

The normal distribution with density function p(x) = (1/2)e(x5)2/2 has mean = 5 and standard deviation = 1.

The integral = (1/2)e(x7)2/2 dx equals 1.

The mean value of a quantity x distributed through a population is the value T such that half of the population has values x less than or equal to T.