Stewart 8th Section 8.5: Probability

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Stewart 8th Section 8.5: Probability 0/26
1 PDF Interpretation · Level 1
Let \(f(x)\) be the probability density function for the lifetime of a manufacturer's highest quality car tire, where \(x\) is measured in miles. Explain the meaning of each integral.
(a) \(\displaystyle\int_{30{,}000}^{40{,}000} f(x) d x\)
(b) \(\displaystyle\int_{25{,}000}^\infty f(x) d x\)

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2 PDF Setup · Level 1
Let \(f(t)\) be the probability density function for the time it takes you to drive to school in the morning, 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

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3 Verify PDF · Level 2
Let \(f(x) = 30 x^2 (1 - x)^2\) for \(0 \leq x \leq 1\) and \(f(x) = 0\) for all other values of \(x\).
(a) Verify that \(f\) is a probability density function.
(b) Find \(P\left(X \leq \dfrac{1}{2}\right)\).

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4 Logistic Distribution · Level 3
The density function \(f(x) = \dfrac{e^{3 - x}}{(1 + e^{3 - x})^2}\) is an example of a logistic distribution.
(a) Verify that \(f\) is a probability density function.
(b) Find \(P(3 \leq X \leq 4)\).
(c) Graph \(f\). What does the mean appear to be? What about the median?

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5 Find Constant in PDF · Level 2
Let \(f(x) = c / (1 + x^2)\).
(a) For what value of \(c\) is \(f\) a probability density function?
(b) For that value of \(c\), find \(P(-1 < X < 1)\).

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6 PDF with Mean · Level 2
Let \(f(x) = k (3 x - x^2)\) if \(0 \leq x \leq 3\) 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.

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7 Uniform Distribution · Level 2
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 \(f(x) = \begin{cases} 0.1 & \quad \text{if } 0 \leq x \leq 10 \\ 0 & \quad \text{if } x < 0 \text{or} x > 10 \end{cases}\) is a probability density function for the spinner's values.
(b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.

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8 PDF from Graph · Level 2
question image
(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) \(P(3 \leq X \leq 8)\)
(c) Calculate the mean.

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9 Median Waiting Time · Level 3
Show that the median waiting time for a phone call to the company described in Example 4 is about \(3.5\) minutes.
10 Exponential Distribution · Level 2
(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?

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11 Exponential Distribution · Level 3
An online retailer has determined that the average time for credit card transactions to be electronically approved is \(1.6\) 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?

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12 PDF Application (Medical) · Level 3
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 \(f(t) = \dfrac{1}{15{,}676} t^2 e^{-0.05 t}\) if \(0 \leq t \leq 150\) 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?

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13 PDF Application (Mean) · Level 3
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 probability density function \(f(t) = \begin{cases} \dfrac{1}{1600} t & \quad \text{if } 0 \leq t \leq 40 \\ \dfrac{1}{20} - \dfrac{1}{1600} t & \quad \text{if } 40 < t \leq 80 \\ 0 \text{otherwise} \end{cases}\), 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.

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14 Normal Distribution · Level 2
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?

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15 Normal Distribution · Level 2
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?
16 Normal Distribution · Level 3
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 can contain less than the stated weight of \(500\) g. At what target weight should the manufacturer set its filling machine?

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17 Normal Distribution · Level 2
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?

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18 Normal Distribution (Proof) · Level 4
Show that the probability density function for a normally distributed random variable has inflection points at \(x = \mu \pm \sigma\).
19 Normal Distribution · Level 3
For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.
20 Standard Deviation · Level 4
The standard deviation for a random variable with probability density function \(f\) and mean \(\mu\) is defined by \(\sigma = [\displaystyle\int_{-\infty}^\infty (x - \mu)^2 f(x) d x]^{\dfrac{1}{2}}\). Find the standard deviation for an exponential density function with mean \(\mu\).
21 Application (Quantum Mechanics) · Level 4
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 \(p(r) = \dfrac{4}{a_0^3} r^2 e^{-2 \dfrac{r}{a_0}}\), \(r \geq 0\), where \(a_0\) is the Bohr radius (\(a_0 \approx 5.59 \times 10^{-11}\) m). The integral \(P(r) = \displaystyle\int_{0}^{r} \dfrac{4}{a_0^3} s^2 e^{-2 \dfrac{s}{a_0}} d s\) gives the probability that the electron will be found within the sphere of radius \(r\) meters centered at the nucleus.
(a) Verify that \(p(r)\) is a probability density function.
(b) Find \(\operatorname*{lim}\limits_{r \rightarrow \infty} p(r)\). 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 \(4 a_0\) centered at the nucleus.
(e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.

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22 Example - Probability Density Function · Level 2
Let \(f(x) = 0.006 x (10 - x)\) for \(0 \leq x \leq 10\) and \(f(x) = 0\) for all other values of \(x\).
(a) Verify that \(f\) is a probability density function.
(b) Find \(P(4 \leq X \leq 8)\).

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23 Example - Exponential Density Function · Level 3
Phenomena such as waiting times and equipment failure times are commonly modeled by exponentially decreasing probability density functions. Find the exact form of such a function.
24 Example - Mean of Exponential Distribution · Level 3
Find the mean of the exponential distribution of Example 2: \(f(t) = \begin{cases} 0 & \quad \text{if } t < 0 \\ c e^{-c t} & \quad \text{if } t \geq 0 \end{cases}\).
25 Example - Exponential Waiting Time · Level 3
Suppose the average waiting time for a customer's call to be answered by a company representative is five minutes.
(a) Find the probability that a call is answered during the first minute, assuming that an exponential distribution is appropriate.
(b) Find the probability that a customer waits more than five minutes to be answered.

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26 Example - Normal Distribution · Level 3
Intelligence Quotient (IQ) scores are distributed normally with mean \(100\) and standard deviation \(15\).
(a) What percentage of the population has an IQ score between \(85\) and \(115\)?
(b) What percentage of the population has an IQ above \(140\)?

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