6.8.1 Consider again the data presented in Section 6.6,
Exercises 1–4, ... Find the median and the mode, and compare
with the expectation. When is the median greater than the expectation?
Experiment a.
Get solution
6.8.2
Consider again the data presented in Section 6.6, Exercises 1–4,
... Find the median and the mode, and compare with the expectation.
When is the median greater than the expectation? Experiment b.
Get solution
6.8.3
Consider again the data presented in Section 6.6, Exercises 1–4,
... Find the median and the mode, and compare with the expectation.
When is the median greater than the expectation? Experiment c.
Get solution
6.8.4
Consider again the data presented in Section 6.6, Exercises 1–4,
... Find the median and the mode, and compare with the expectation.
When is the median greater than the expectation? Experiment d.
Get solution
6.8.5 Using the histograms (from Section 6.6, Exercises 9–12), estimate the median, the mode, and the expectation. ...
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6.8.6 Using the histograms (from Section 6.6, Exercises 9–12), estimate the median, the mode, and the expectation. ...
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6.8.7 Using the histograms (from Section 6.6, Exercises 9–12), estimate the median, the mode, and the expectation. ...
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6.8.8 Using the histograms (from Section 6.6, Exercises 9–12), estimate the median, the mode, and the expectation. ...
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6.8.9 Find
the median and mode of the continuous random variables with the given
p.d.f., and compare with the expectation as found earlier. The p.d.
f. of a random variable X is f (x)=2x for 0 ≤ x ≤1 (as in Section 6.6, Exercises 17 and 21, and Section 6.7,Exercise 7).
Get solution
6.8.10 Find
the median and mode of the continuous random variables with the given
p.d.f., and compare with the expectation as found earlier. The p.d.
f. of a random variable X is ...for 0≤ x ≤2 (as in Section 6.6, Exercises 18 and 22, and Section 6.7, Exercise 8).
Get solution
6.8.11 Find
the median and mode of the continuous random variables with the given
p.d.f., and compare with the expectation as found earlier. The p.d.
f. of a random variable T is ...for 1≤t ≤e (as in Section 6.6, Exercises 19 and 23, and Section 6.7, Exercise 9).
Get solution
6.8.12 Find
the median and mode of the continuous random variables with the given
p.d.f., and compare with the expectation as found earlier. The p.d.
f. of a random variable T is g(t)=6t (1 − t) for 0≤ t ≤1 (as in Section 6.6, Exercises 20 and 24, and Section 6.7, Exercise 10).
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6.8.13 Recall
the following slightly peculiar p.d.f.’s from Section 6.7, Exercises 15
and 16. Find the median of each. How does it compare with the
expectation? ... for 0 < x ≤1 (from Section 6.7, Exercise 15).
Get solution
6.8.14 Recall
the following slightly peculiar p.d.f.’s from Section 6.7, Exercises 15
and 16. Find the median of each. How does it compare with the
expectation? ... for 1 ≤ t <∞ (from Section 6.7, Exercise 16).
Get solution
6.8.15
Find the arithmetic and geometric means of the following random
variables. Check that the arithmetic-geometric inequality holds. The
random variable X where Pr(X =1)=0.3 and Pr(X = 2)=0.7.
Get solution
6.8.16
Find the arithmetic and geometric means of the following random
variables. Check that the arithmetic-geometric inequality holds. The
random variable X where Pr(X =1)=0.4 and Pr(X = 3)=0.6.
Get solution
6.8.17
Find the arithmetic and geometric means of the following random
variables. Check that the arithmetic-geometric inequality holds. The
random variable X where Pr(X =1)=0.3, Pr(X =2)= 0.3, and Pr(X =3)=0.4.
Get solution
6.8.18
Find the arithmetic and geometric means of the following random
variables. Check that the arithmetic-geometric inequality holds. The
random variable X where Pr(X =2)=0.1, Pr(X =3)= 0.2, and Pr(X =5)=0.7.
Get solution
6.8.19 Find the geometric mean of the continuous random variables with the given p.d.
f. Compare with the expectation. The p.d.
f. of a random variable X is f (x)=2 for 0.75 ≤ x ≤ 1.25. Use Equation 6.8.2. Equation 6.8.2 ...
Get solution
6.8.20 Find the geometric mean of the continuous random variables with the given p.d.
f. Compare with the expectation. The p.d.
f. of a random variable X is f (x)=5 for 1 ≤ x ≤1.2.
Get solution
6.8.21 Find the geometric mean of the continuous random variables with the given p.d.
f. Compare with the expectation. The p.d.
f. of a random variable X is f (x)=2x for 0 ≤ x ≤1
(as in Section 6.6, Exercise 17 and Section 6.7, Exercise 7). Use
Equation 6.8.3, and you will have to use L’Hˆopital’s rule to evaluate
the integral. Equation 6.8.3 ...
Get solution
6.8.22 Find the geometric mean of the continuous random variables with the given p.d.
f. Compare with the expectation. The p.d.
f. of a random variable T is ...for 1≤t ≤e (as in Section 6.6, Exercise 19 and Section 6.7, Exercise 9). HINT: Use the substitution u = ln(t) to do the integral.
Get solution
6.8.23 The geometry behind the geometric mean is based on the following argument. If a random variable R takes on each of the values ...with probability 0.5, a rectangle with sides of length ...has area equal to that of a square with sides with length equal to the geometric mean. Check this in the case that ...
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6.8.24 Check this in general, without picking values for ...
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6.8.25 Fix ... =1. Find the value of ... that maximizes the ratio of the geometric mean to the arithmetic mean.
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6.8.26 Prove that the geometric mean is always less than or equal to the arithmetic mean (the arithmetic-geometric inequality) when ... =1 (the case described in Exercise 25).
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6.8.27 Suppose that incomes in a company have the following probabilities. ... For
the given values of the top salary, find the mean, median, and mode,
and say which statistic is most informative about the distribution of
salaries. (This is based on an example in the book How to Lie with Statistics.) The top salary is $450,000.
Get solution
6.8.28 Suppose that incomes in a company have the following probabilities. ... For
the given values of the top salary, find the mean, median, and mode,
and say which statistic is most informative about the distribution of
salaries. (This is based on an example in the book How to Lie with Statistics.) The top salary is $4,500,000.
Get solution
6.8.29 For the data (from Section 6.6, Exercises 37–40), find the median and the mode. ... Experiment a
Get solution
6.8.30 For the data (from Section 6.6, Exercises 37–40), find the median and the mode. ... Experiment b
Get solution
6.8.31 For the data (from Section 6.6, Exercises 37–40), find the median and the mode. ... Experiment c
Get solution
6.8.32 For the data (from Section 6.6, Exercises 37–40), find the median and the mode. ... Experiment d
Get solution
6.8.33 As in Section 6.6, Exercises 45 and 46, the p.d.
f. for the waiting time X until an event occurs often follows the exponential distribution, with the form ...for some positive value of λ, defined for x ≥0. Find the median waiting time for the following values of λ and compare with the expectation (found in Section 6.7, Exercises 35 and 36). λ=0.5.
Get solution
6.8.34 As in Section 6.6, Exercises 45 and 46, the p.d.
f. for the waiting time X until an event occurs often follows the exponential distribution, with the form ...for some positive value of λ, defined for x ≥0. Find the median waiting time for the following values of λ and compare with the expectation (found in Section 6.7, Exercises 35 and 36). λ=2.0.
Get solution
6.8.35 The
following problems show that most people live in places that are more
crowded than average. In each case, find the average size of a city, the
average crowding a person experiences, and the fraction of people who
live in places more crowded than average. There are two cities, one with 100,000 people and the other with 1,000,000 people.
Get solution
6.8.36
The
following problems show that most people live in places that are more
crowded than average. In each case, find the average size of a city, the
average crowding a person experiences, and the fraction of people who
live in places more crowded than average. There are three cities, one
with 100,000 people, one with 400,000 people, and the other with
1,000,000 people.
Get solution
6.8.37 Find the arithmetic and geometric means of the random variables R for
per capita production in the following cases. Check that the
arithmetic-geometric inequality holds in each case. Which describes a
growing population? R =4 with probability 0.5, R =0.25 with probability 0.5.
Get solution
6.8.38 Find the arithmetic and geometric means of the random variables R for
per capita production in the following cases. Check that the
arithmetic-geometric inequality holds in each case. Which describes a
growing population? R =4 with probability 0.25, R =0.25 with probability 0.75.
Get solution
6.8.39 Find the arithmetic and geometric means of the random variables R for
per capita production in the following cases. Check that the
arithmetic-geometric inequality holds in each case. Which describes a
growing population? R =4 with probability 0.75, R =0.25 with probability 0.25.
Get solution
6.8.40 Find the arithmetic and geometric means of the random variables R for
per capita production in the following cases. Check that the
arithmetic-geometric inequality holds in each case. Which describes a
growing population? R =5 with probability 0.25, R =0.25 with probability 0.25, R =1 with probability 0.5.
Get solution
6.8.41
Suppose populations start at 100. Estimate the population size after
50 generations in the following cases. The situation in Exercise 37.
Exercise 37 R =4 with probability 0.5, R =0.25 with probability 0.5.
Get solution
6.8.42
Suppose populations start at 100. Estimate the population size after
50 generations in the following cases. The situation in Exercise 38.
Exercise 38 R =4 with probability 0.25, R =0.25 with probability 0.75.
Get solution
6.8.43
Suppose populations start at 100. Estimate the population size after
50 generations in the following cases. The situation in Exercise 39.
Exercise 39 =4 with probability 0.75, R =0.25 with probability 0.25.
Get solution
6.8.44
Suppose populations start at 100. Estimate the population size after
50 generations in the following cases. The situation in Exercise 40.
Exercise 40 R =5 with probability 0.25, R =0.25 with probability 0.25, R =1 with probability 0.5.
Get solution
6.8.45 A
store has two managers, one who believes that high profits come from
lowering prices and getting more customers and another who believes that
high profits come from raising prices and making more profit per
customer. These managers get to choose prices in alternate weeks. For
each case,
a. Find the price after 1, 2, 3, and 4 weeks.
b. Find a formula for the price after t weeks (break into the two cases, t even and t odd).
c. Which of the managers wins?
d. What does this have to do with the geometric mean? In
week 1, the “low price” manager cuts prices by 50%. In week 2, the
“high price” manager raises prices by 50%,and so forth. Suppose an item
started out at $100.
Get solution
6.8.46 A
store has two managers, one who believes that high profits come from
lowering prices and getting more customers and another who believes that
high profits come from raising prices and making more profit per
customer. These managers get to choose prices in alternate weeks. For
each case,
a. Find the price after 1, 2, 3, and 4 weeks.
b. Find a formula for the price after t weeks (break into the two cases, t even and t odd).
c. Which of the managers wins?
d. What does this have to do with the geometric mean? In
week 1, the “low price” manager cuts prices by 20%. In week 2, the
“high price” manager raises prices by 30%, and so forth. Suppose an item
started out at $100.
Get solution
6.8.47 The p.d.
f. for a random variable taking on values between 0.8 and 1.1 with equal probability is f (x)=10/3 for 0.8≤ x ≤1.1. Find the geometric mean r of this random variable. Define two updating functions, a deterministic g for a population that has per capita production of exactly r and a stochastic G. Compare the dynamics of the two populations for 100 steps starting from an initial condition of 100. How similar do they look?
Get solution
6.8.48 Suppose
that the per capita production of a population is a random variable
with uniform (flat) p.d.
f. on the interval from 0.8 to y, where y is an unknown value. To guarantee that the integral is equal to 1, the height of the p.d.
f. must be ...rather than 3.333. Why is this?
a. Compute a function H(y), which gives the geometric mean as a function of y.
b. Solve for the value ... for which the geometric mean is 1.0.
c. Define a random variable R to produce random numbers in the interval from 0.8 to ... and an associated updating function G to describe a population with per capita production equal to R.
d. Generate two trajectories of 100 generations using the updating function G starting from N =100.
e. Where do you expect the trajectories to end up? How close are they?
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