Solved: The speeds of vehicles on a highway with speed limit are normally distributed

Let be the probability density function for the lifetime of a manufacturers highest quality car tire, where is measured in miles. Explain the meaning of each integral.y 40,000 30,000 f x dx y  25,000 f x dx f t

Let be the probability density function for the time it takes you to drive to school in the morning, where is measured in minutes. Express the following probabilities as integrals. (a) The probability that you drive to school in less than 15 minutes (b) The probability that it takes you more than half an hour to get to school

Let for and for all other values of . (a) Verify that is a probability density function. (b) Find .

Let if and if . (a) Verify that is a probability density function. (b) Find .

Let . (a) For what value of is a probability density function? (b) For that value of , find .

Let if and if or . (a) For what value of is a probability density function? (b) For that value of , find . (c) Find the mean.

A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length. (a) Explain why the function is a probability density function for the spinners values. (b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.

. (a) Explain why the function whose graph is shown is a probability density function. (b) Use the graph to find the following probabilities: (i) (ii) (c) Calculate the mean.

Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.

(a) A type of lightbulb is labeled as having an average lifetime of 1000 hours. Its reasonable to model the probability of failure of these bulbs by an exponential density function with mean . Use this model to find the probability that a bulb (i) fails within the first 200 hours, (ii) burns for more than 800 hours. (b) What is the median lifetime of these lightbulbs?

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 minutes. (a) Find the probability that a customer has to wait more than 4 minutes. (b) Find the probability that a customer is served within the first 2 minutes. (c) The manager wants to advertise that anybody who isnt served within a certain number of minutes gets a free hamburger. But she doesnt want to give away free hamburgers to more than 2% of her customers. What should the advertisement say?

According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall? (b) What percentage of the adult male population is more than 6 feet tall?

The Garbage Project at the University of Arizona reports that the amount of paper discarded by households per week is normally distributed with mean 9.4 lb and standard deviation 4.2 lb. What percentage of households throw out at least 10 lb of paper a week?

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are nor mally distributed with standard deviation 12 g. (a) If the target weight is 500 g, what is the probability that the machine produces a box with less than 480 g of cereal? (b) Suppose a law states that no more than 5% of a manufacturers cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine?

The speeds of vehicles on a highway with speed limit are normally distributed with mean and standard deviation . (a) What is the probability that a randomly chosen vehicle is traveling at a legal speed? (b) If police are instructed to ticket motorists driving or more, what percentage of motorists are targeted?

Show that the probability density function for a normally distributed random variable has inflection points at .

For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.

The standard deviation for a random variable with probability density function and mean is defined by Find the standard deviation for an exponential density function with mean .

The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a cloud of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or 1s-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function where is the Bohr radius . The integral gives the probability that the electron will be found within the sphere of radius meters centered at the nucleus. (a) Verify that is a probability density function. (b) Find . For what value of does have its maximum value? ; (c) Graph the density function. (d) Find the probability that the electron will be within the sphere of radius centered at the nucleus. (e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.

. Lengths of human pregnancies are normally distributed with mean 268 days and standard deviation 15 days. What per cen tage of pregnancies last between 250 days and 280 days?

The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes. (a) What is the probability that a customer is served in the first 3 minutes? (b) What is the probability that a customer has to wait more than 10 minutes? (c) What is the median waiting time?