?Let \(f(x)=30 x^{2}(1-x)^{2}\) for \(0 \leqslant x \leqslant 1\) and \(f(x)=0\) for all

Let \(f(x)\) be the probability density function for the lifetime of a manufacturer's highest quality car tire, where is measured in miles. Explain the meaning of each integral. (a) \(\int_{30,000}^{40,000} f(x) d x\) (b) \(\int_{25,000}^{\infty} f(x) d x\) Equation Transcription: Text Transcription: f(x) integral _30,000 ^40,000 f(x) dx integral _25,000 ^infinity f(x) dx

Let \(f(t)\) be the probability density function for the time it takes you to drive to school, where \(t\) 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 Equation Transcription: Text Transcription: f(t) t

Let \(f(x)=30 x^{2}(1-x)^{2}\) for \(0 \leqslant x \leqslant 1\) and \(f(x)=0\) for all other values of \(x\). (a) Verify that \(f\) is a probability density function. (b) Find \(P\left(X \leqslant \frac{1}{3}\right)\). Equation Transcription: ? ? ? Text Transcription: f(x)=30x^2(1-x)^2 0 leqslant x leqslant 1 f(x)=0 x f P(X leqslant 1/3)

The density function \(f(x)=\frac{e^{3-x}}{\left(1+e^{3-x}\right)^{2}}\) is an example of a logistic distribution. (a) Verify that \(f\) is a probability density function. (b) Find \(P(3 \leqslant X \leqslant 4)\). (c) Graph \(f\). What does the mean appear to be? What about the median? Equation Transcription: ? ? Text Transcription: f(x)=e^3-x/(1+e^3-x)^2 f P(3 leqslant X leqslant 4) f

Let \(f(x)=c /\left(1+x^{2}\right)\). (a) For what value of \(c\) is \(f\) a probability density function? (b) For that value of \(c\), find \(P(-1<X<1)\). Equation Transcription: . Text Transcription: f(x)=c/(1+x^2). c f c P(-1<X<1)

Let \(f(x)=k\left(3 x-x^{2}\right)\) if \(0 \leqslant x \leqslant 3\) and \(f(x)=0\) if \(x<0\) or \(x>3\). (a) For what value of k is f a probability density function? (b) For that value of k, find P(X > 1). (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 spinner's values. \(f(x)= \begin{cases}0.1 & \text { if } 0 \leqslant x \leqslant 10 \\ 0 & \text { if } x<0 \text { or } x>10\end{cases}\) (b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral. ________________ Equation Transcription: Text Transcription: f(x) = 0.1 & { if } 0 leqslant x leqslant 10 \\ 0 & { if } x < 0 { or } x > 10

(a) Explain why the function whose graph is shown is a probability density function. (b) Use the graph to find the following probabilities: (i) \(P(X<3)\) (ii) \(P(3 \leqslant X \leqslant 8)\) (c) Calculate the mean. ________________ Equation Transcription: ? ? Text Transcription: P (X < 3) P(3 leqslant X leqslant 8)

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

(a) A type of light bulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean \(\mu=1000\). 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 light bulbs? ________________ Equation Transcription: Text Transcription: mu = 1000

An online retailer has determined that the average time for credit card transactions to be electronically approved is seconds. (a) Use an exponential density function to find the probability that a customer waits less than a second for credit card approval. (b) Find the probability that a customer waits more than 3 seconds. (c) What is the minimum approval time for the slowest 5% of transactions?

The time between infection and the display of symptoms for streptococcal sore throat is a random variable whose probability density function can be approximated by \(f(t)=\frac{i}{15,676} t^{2} e^{-0.05 t}\) if \(0 \leqslant t \leqslant 150\) and \(f(t)=0\) otherwise ( \(t\) measured in hours). (a) What is the probability that an infected patient will display symptoms within the first 48 hours? (b) What is the probability that an infected patient will not display symptoms until after 36 hours? Source: Adapted from P. Sartwell, "The Distribution of Incubation Periods of Infectious Disease," American Journal of Epidemiology 141 (1995): 386-94.

REM sleep is the phase of sleep when most active dreaming occurs. In a study, the amount of REM sleep during the first four hours of sleep was described by a random variable \(T\) with a probability density function \(f(t)= \begin{cases}\frac{1}{1600} t & \text { if } 0 \leqslant t \leqslant 40 \\frac{1}{20}-\frac{1}{1600} t & \text { if } 40<t \leqslant 80 \\ 0 & \text { otherwise }\end{cases}\) where \(t\) is measured in minutes. (a) What is the probability that the amount of REM sleep is between 30 and 60 minutes? (b) Find the mean amount of REM sleep. ________________ Equation Transcription: { Text Transcription: T f(t) = {frac{1}{1600} t & { if } 0 leqslant t leqslant 40 \\ \frac{1}{20} - frac{1}{1600} t & { if } 40 < t leqslant 80 \\ 0 & otherwise t

According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean inches and standard deviation 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 \(10lb\) of paper a week? ________________ Equation Transcription: Text Transcription: 9.4 lb 4.2 lb 10 lb

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally 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 manufacturer's cereal boxes may 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 100 km/h are normally distributed with mean 112 km/h and standard deviation 8 km/h. (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 125 km/h or more, what percentage of motorists are targeted?

Show that the probability density function for a normally distributed random variable has inflection points at \(x=\mu \pm \sigma\). ________________ Equation Transcription: Text Transcription: x = mu pm sigma

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 \(f\) and mean is defined by \(\sigma=\left[\int_{-\infty}^{\infty}(x-\mu)^{2} f(x) d x\right]^{1 / 2}\) Find the standard deviation for an exponential density function with mean \(\mu\). Equation Transcription: Text Transcription: F mu Sigma = [integral_ -infinity ^infinity (x-mu)^2 f(x) dx]^½ mu

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 -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 \(p(r)=\frac{4}{a_{0}^{3}} r^{2} e^{-2 r / a_{0}} \quad r \geq 0\) where \( is the Bohr radius \(p(r)=\frac{4}{a_{0}^{3}} r^{2} e^{-2 r / a_{0}} r \geq 0\).The integral \(P(r)=\int_{0}^{r} \frac{4}{a_{0}^{s}} s^{2} e^{-2 s / a_{0}} d s\) gives the probability that the electron will be found within the sphere of radius meters centered at the nucleus. (a) Verify that \(p(r)\) is a probability density function. (b) Find \(\lim _{r \rightarrow \infty} p(r)\). for what value of \(r\) does \(p(r)\) have its maximum value? (c) Graph the density function. (d) Find the probability that the electron will be within the sphere of radius \(4a_0\) centered at the nucleus. (e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom. Equation Transcription: Text Transcription: p(r) =4/a sub 0 ^3 r^2 e^-2r/asub 0 r> or = 0 (a. sub 0 almost equals 5.59 x 10^- 11 m) P(r) =0r4a03s2e-2s/a0ds p(r) limrrightarrowinfinityp(r) r p(r) 4a0