A congressional committee is investigating a defense contractor whose projects often

(a) Using the density function in Example 2 on page 334, fill in values for the cumulative distribution function P(t) for the length of time people wait in the doctors office. t (hours) 0 1 2 3 4 P(t) (fraction of people waiting) (b) Graph P(t).

Show that the area under the fishing density function in Figure 7.12 on page 339 is 1. Why is this to be expected?

Figure 7.15 shows a density function and the corresponding cumulative distribution function.5 (a) Which curve represents the density function and which represents the cumulative distribution function? Give a reason for your choice. (b) Put reasonable values on the tick marks on each of the axes. Figure 7.15

In an agricultural experiment, the quantity of grain from a given size field ismeasured. The yield can be anything from 0 kg to 50 kg. For each of the following situations, pick the graph that best represents the: (i) Probability density function (ii) Cumulative distribution function. (a) Low yields are more likely than high yields. (b) All yields are equally likely. (c) High yields are more likely than low yields.

Decide if the function graphed in Problems 510 is a probability density function (pdf) or a cumulative distribution function (cdf). Give reasons. Find the value of c. Sketch and label the other function. (That is, sketch and label the cdf if the problem shows a pdf, and the pdf if the problem shows a cdf.)

Decide if the function graphed in Problems 510 is a probability density function (pdf) or a cumulative distribution function (cdf). Give reasons. Find the value of c. Sketch and label the other function. (That is, sketch and label the cdf if the problem shows a pdf, and the pdf if the problem shows a cdf.)

Decide if the function graphed in Problems 510 is a probability density function (pdf) or a cumulative distribution function (cdf). Give reasons. Find the value of c. Sketch and label the other function. (That is, sketch and label the cdf if the problem shows a pdf, and the pdf if the problem shows a cdf.)

Decide if the function graphed in Problems 510 is a probability density function (pdf) or a cumulative distribution function (cdf). Give reasons. Find the value of c. Sketch and label the other function. (That is, sketch and label the cdf if the problem shows a pdf, and the pdf if the problem shows a cdf.)

Decide if the function graphed in Problems 510 is a probability density function (pdf) or a cumulative distribution function (cdf). Give reasons. Find the value of c. Sketch and label the other function. (That is, sketch and label the cdf if the problem shows a pdf, and the pdf if the problem shows a cdf.)

Decide if the function graphed in Problems 510 is a probability density function (pdf) or a cumulative distribution function (cdf). Give reasons. Find the value of c. Sketch and label the other function. (That is, sketch and label the cdf if the problem shows a pdf, and the pdf if the problem shows a cdf.)

A person who travels regularly on the 9:00 am bus from Oakland to San Francisco reports that the bus is almost always a few minutes late but rarely more than five minutes late. The bus is never more than two minutes early, although it is on very rare occasions a little early. (a) Sketch a density function, p(t), where t is the number of minutes that the bus is late. Shade the region under the graph between t = 2 minutes and t = 4 minutes. Explain what this region represents. (b) Now sketch the cumulative distribution function P(t). What measurement(s) on this graph correspond to the area shaded? What do the inflection point(s) on your graph of P correspond to on the graph of p? Interpret the inflection points on the graph of P without referring to the graph of p.

Suppose F(x) is the cumulative distribution function for heights (in meters) of trees in a forest. (a) Explain in terms of trees the meaning of the statement F(7) = 0.6. (b) Which is greater, F(6) or F(7)? Justify your answer in terms of trees.

Students at the University of California were surveyed and asked their grade point average. (The GPA ranges from 0 to 4, where 2 is just passing.) The distribution of GPAs is shown in Figure 7.16.6 (a) Roughly what fraction of students are passing? (b) Roughly what fraction of the students have honor grades (GPAs above 3)? (c) Why do you think there is a peak around 2? (d) Sketch the cumulative distribution function.

The density function and cumulative distribution function of heights of grass plants in a meadow are in Figures 7.17 and 7.18, respectively. (a) There are two species of grass in the meadow, a short grass and a tall grass. Explain how the graph of the density function reflects this fact. (b) Explain how the graph of the cumulative distribution function reflects the fact that there are two species of grass in the meadow. (c) About what percentage of the grasses in the meadow belong to the short grass species? 0.5 1 1.5 2 height (meter) fraction of plants per meter of height Figure 7.17 0.5 1 1.5 2 0.25 0.5 0.75 1 height (meter) fraction of plants Figure 7.18

A congressional committee is investigating a defense contractor whose projects often incur cost overruns. The data in Table 7.7 show y, the fraction of the projects with an overrun of at most C%. (a) Plot the data with C on the horizontal axis. Is this a density function or a cumulative distribution function? Sketch a curve through these points. (b) If you think you drew a density function in part (a), sketch the corresponding cumulative distribution function on another set of axes. If you think you drew a cumulative distribution function in part (a), sketch the corresponding density function. (c) Based on the table, what is the probability that there will be a cost overrun of 50% or more? Between 20% and 50%? Nearwhat percent is the cost overrun most likely to be? Table 7.7 Fraction, y, of overruns that are at most C% C 20% 10% 0% 10% 20% 30% 40% 50% y 0.01 0.08 0.19 0.32 0.50 0.80 0.94 0.99

Find the probability that a banana will last (a) Between 1 and 2 weeks. (b) More than 3 weeks. (c) More than 4 weeks.

(a) Sketch the cumulative distribution function for the shelf life of bananas. [Note: The domain of your function should be all real numbers, including to the left of t = 0 and to the right of t = 4.] (b) Use the cumulative distribution function to estimate the probability that a banana lasts between 1 and 2 weeks. Check with Problem 16(a).

An experiment is done to determine the effect of two new fertilizers A and B on the growth of a species of peas. The cumulative distribution functions of the heights of the mature peas without treatment and treated with each of A and B are graphed in Figure 7.20. (a) About what height are most of the unfertilized plants? (b) Explain in words the effect of the fertilizers A and B on the mature height of the plants. 1 2 A B 1 x height (meters) fraction of plants Unfertilized Figure 7.20

A group of people have received treatment for cancer. Let t be the survival time, the number of years a person lives after the treatment. The density function giving the distribution of t is p(t) = CeCt for some positive constant C. What is the practicalmeaning of the cumulative distribution function P(t) = =t 0 p(x) dx?

The probability of a transistor failing between t = a months and t = b months is given by c =b a ectdt, for some constant c. (a) If the probability of failure within the first six months is 10%, what is c? (b) Given the value of c in part (a), what is the probability the transistor fails within the second six months?

While taking a walk along the road where you live, you accidentally drop your glove, but you dont know where. The probability density p(x) for having dropped the glove x kilometers from home (along the road) is p(x) = 2e2x for x 0. (a) What is the probability that you dropped it within 1 kilometer of home? (b) At what distance y from home is the probability that you dropped it within y km of home equal to 0.95?