Chapter 7

A quantity \(x\) is distributed with density function \(p(x)=\) \(0.5(2-x)\) for \(0 \leq x \leq 2\) and \(p(x)=0\) otherwise. Find the mean and median of \(x\)

In an agricultural experiment, the quantity of grain from a given size field is measured. The yield can be anything from \(0 \mathrm{kg}\) to \(50 \mathrm{kg}\). 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. (GRAPHS CAN'T COPY)

A quantity \(x\) has cumulative distribution function \(P(x)=x-x^{2} / 4\) for \(0 \leq x \leq 2\) and \(P(x)=0\) for \(x<0\) and \(P(x)=1\) for \(x>2 .\) Find the mean and median of \(x\)

Let \(p(t)=-0.0375 t^{2}+0.225 t\) be the density function for the shelf life of a brand of banana which lasts up to 4 weeks. Time, \(t\), is measured in weeks and \(0 \leq t \leq 4\) Find the median shelf life of a banana using \(p(t) .\) Plot the median on a graph of \(p(t) .\) Does it look like half the area is to the right of the median and half the area is to the left?

Suppose that \(x\) measures the time (in hours) it takes for a student to complete an exam. All students are done within two hours and the density function for \(x\) is $$ p(x)=\left\\{\begin{array}{ll} x^{3} / 4 & \text { if } 0

Let \(p(x)\) be the density function for annual family income, where \(x\) is in thousands of dollars. What is the meaning of the statement \(p(70)=0.05 ?\)

Let \(p(t)=0.1 e^{-0.1 t}\) be the density function for the waiting time at a subway stop, with \(t\) in minutes, \(0 \leq t \leq 60\) (a) Graph \(p(t) .\) Use the graph to estimate visually the median and the mean. (b) Calculate the median and the mean. Plot both on the graph of \(p(t)\) (c) Interpret the median and mean in terms of waiting time.

The speeds of cars on a road are approximately normally distributed with a mean \(\mu=58 \mathrm{km} / \mathrm{hr}\) and standard deviation \(\sigma=4 \mathrm{km} / \mathrm{hr}\) (a) What is the probability that a randomly selected car is going between 60 and \(65 \mathrm{km} / \mathrm{hr} ?\) (b) What fraction of all cars are going slower than 52 \(\mathrm{km} / \mathrm{hr} ?\)

The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation 15 (a) Write the formula for the density function of IQ scores. (b) Estimate the fraction of the population with IQ between 115 and 120

Let \(P(x)\) be the cumulative distribution function for the household income distribution in the US in \(2009 .^{8}\) Values of \(P(x)\) are in the following table: $$\begin{array}{l|c|c|c|c|c}\hline \text { Income } x \text { (thousand } \$ \text { ) } & 20 & 40 & 60 & 75 & 100 \\\\\hline P(x)(\%) & 29.5 & 50.1 & 66.8 & 76.2 & 87.1 \\\\\hline\end{array}$$ (a) What percent of the households made between \(\$ 40,000\) and \(\$ 60,000 ?\) More than \(\$ 100,000 ?\) (b) Approximately what was the median income? (c) Is the statement "More than one-third of households made between \(\$ 40,000\) and \(\$ 75,000\) " true or false?

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.

Find a density function \(p(x)\) such that \(p(x)=0\) when \(x \geq 5\) and when \(x<0,\) and is decreasing when \(0 \leq x \leq 5\)

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 \(50 \%\) ? Near what percent is the cost overrun most likely to be? Fraction, \(y,\) of overruns that are at most \(C \%\) $$\begin{array}{c|c|c|c|c|c|c|c|c}\hline C & -20 \mathrm{s} & -10 \% & 0 \% & 10 \% & 20\mathrm{se} & 30 \mathrm{se} & 40 \mathrm{se} & 50 \mathrm{st} \\\\\hline y & 0.01 & 0.08 & 0.19 & 0.32 & 0.50 & 0.80 & 0.94 & 0.99 \\\\\hline\end{array}$$

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. All yields from 0 to 100 kg are equally likely; the field never yields more than \(100 \mathrm{kg}\).

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. High yields are more likely than low. The maximum yield is \(200 \mathrm{kg}\)

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. A drought makes low yields most common, and there is no yield greater than \(30 \mathrm{kg}\).

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 \(p(t)=C e^{-C t}\) for some positive constant \(C .\) What is the practical meaning of the cumulative distribution function \(P(t)=\int_{0}^{t} p(x) d x ?\)

The probability of a transistor failing between \(t=a\) months and \(t=b\) months is given by \(c \int_{a}^{b} e^{-c t} d t,\) 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 don't know where. The probability density \(p(x)\) for having dropped the glove \(x\) kilometers from home (along the road) is $$p(x)=2 e^{-2 x} \quad \text { for } x \geq 0$$ (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 \(0.95 ?\)