6.9.1 For
the given data (first presented in Section 6.6, Exercises 1– 4), find
the range, MAD, the variance both the direct way and with the
computational formula, the standard deviation, and the coefficient of
variation. ... Experiment a.
Get solution
6.9.2 For
the given data (first presented in Section 6.6, Exercises 1– 4), find
the range, MAD, the variance both the direct way and with the
computational formula, the standard deviation, and the coefficient of
variation. ... Experiment b.
Get solution
6.9.3 For
the given data (first presented in Section 6.6, Exercises 1– 4), find
the range, MAD, the variance both the direct way and with the
computational formula, the standard deviation, and the coefficient of
variation. ... Experiment c.
Get solution
6.9.4 For
the given data (first presented in Section 6.6, Exercises 1– 4), find
the range, MAD, the variance both the direct way and with the
computational formula, the standard deviation, and the coefficient of
variation. ... Experiment d.
Get solution
6.9.5 Consider
the following random variables that take only two values with the given
probabilities. For each, find MAD, the variance, the standard
deviation, and the coefficient of variation. A Bernoulli random variable with p =1/3.
Get solution
6.9.6 Consider
the following random variables that take only two values with the given
probabilities. For each, find MAD, the variance, the standard
deviation, and the coefficient of variation. A Bernoulli random variable with p =0.9.
Get solution
6.9.7 Consider
the following random variables that take only two values with the given
probabilities. For each, find MAD, the variance, the standard
deviation, and the coefficient of variation. A random variable that takes the value 10 with probability 1/3 and the value of 0 with probability 2/3. Compare your answers with the answer to Exercise 5.
Get solution
6.9.8 Consider
the following random variables that take only two values with the given
probabilities. For each, find MAD, the variance, the standard
deviation, and the coefficient of variation. A random variable that takes the value 11 with probability 0.9 and the value of 10 with probability 0.1. Compare your answers with the answer to Exercise 6.
Get solution
6.9.9 Find the quartiles of a random variable with the given p.d.
f. Illustrate the areas on a graph of the p.d.
f. 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).
Get solution
6.9.10 Find the quartiles of a random variable with the given p.d.
f. Illustrate the areas on a graph of the p.d.
f. The p.d.
f. of a random variable X is ...for 0 ≤ x ≤2 (as in Section 6.6, Exercise 18 and Section 6.7, Exercise 8).
Get solution
6.9.11 Find the quartiles of a random variable with the given p.d.
f. Illustrate the areas on a graph of the p.d.
f. 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).
Get solution
6.9.12 Find the quartiles of a random variable with the given p.d.
f. Illustrate the areas on a graph of the p.d.
f. The p.d.
f. of a random variable T is g(t)=6t (1 − t) for 0≤t ≤1
(as in Section 6.6, Exercise 20 and Section 6.7, Exercise 10). This
requires a computer (or Newton’s method) to solve the equations.
Get solution
6.9.13 Find the variance and standard deviation of a continuous random variable with the given p.d.
f. 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).
Get solution
6.9.14 Find the variance and standard deviation of a continuous random variable with the given p.d.
f. The p.d.
f. of a random variable X is ...for 0 ≤ x ≤2 (as in Section 6.6, Exercise 18 and Section 6.7, Exercise 8).
Get solution
6.9.15 Find the variance and standard deviation of a continuous random variable with the given p.d.
f. 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).
Get solution
6.9.16 Find the variance and standard deviation of a continuous random variable with the given p.d.
f. The p.d.
f. of a random variable T is g(t)=6t (1 − t) for 0 ≤ t ≤1 (as in Section 6.6, Exercise 20 and Section 6.7, Exercise 10).
Get solution
6.9.17 Find
MAD for a continuous random variable with the given p.d.
f. How does it
compare with the standard deviation found in the earlier problem? The p.d.
f. of a random variable X is f (x)=2x for 0 ≤ x ≤1 (as in Exercise 13).
Get solution
6.9.18 Find
MAD for a continuous random variable with the given p.d.
f. How does it
compare with the standard deviation found in the earlier problem? The p.d.
f. of a random variable X is ...for 0 ≤ x ≤2 (as in Exercise 14).
Get solution
6.9.19 Find
the probability that the random variable has a value less than one
standard deviation below the expectation and less than two standard
deviations below the expectation. How do the results compare with the
rules of thumb for a bell-shaped distribution? The p.d.
f. of a random variable X is f (x)=2x for 0 ≤ x ≤1 (as in Exercise 13).
Get solution
6.9.20 Find
the probability that the random variable has a value less than one
standard deviation below the expectation and less than two standard
deviations below the expectation. How do the results compare with the
rules of thumb for a bell-shaped distribution? The p.d.
f. of a random variable X is ...for 0 ≤ x ≤2 (as in Exercise 14).
Get solution
6.9.21 Find
the probability that the random variable has a value less than one
standard deviation below the expectation and less than two standard
deviations below the expectation. How do the results compare with the
rules of thumb for a bell-shaped distribution? The p.d.
f. of a random variable T is ...for 1≤t ≤e (as in Exercise 15).
Get solution
6.9.22 Find
the probability that the random variable has a value less than one
standard deviation below the expectation and less than two standard
deviations below the expectation. How do the results compare with the
rules of thumb for a bell-shaped distribution? The p.d.
f. of a random variable T is g(t)=6t (1 − t) for 0 ≤ t ≤1 (as in Exercise 16).
Get solution
6.9.23
The following steps outline the proof of the computational formula
for the variance. Multiply out the squared term into three terms and
break the sum into three sums.
Get solution
6.9.24
The following steps outline the proof of the computational formula
for the variance. To further simplify, try the following steps.
a. Factor constants out of the sums.
b. Remember that ...is a constant.
c. Recognize certain sums to be equal to the mean.
d. Write in terms of the mean and group together like terms.
Get solution
6.9.25 There is a general inequality about any random variable X, called Chebyshev’s inequality. Suppose X has mean μ and standard deviation σ. Then ... for any value of k. What does this mean for k =1? Does this tell us anything?
Get solution
6.9.26 There is a general inequality about any random variable X, called Chebyshev’s inequality. Suppose X has mean μ and standard deviation σ. Then ... for any value of k. What does this mean for k =2? How much of the probability must lie within two standard deviations of the mean?
Get solution
6.9.27 There is a general inequality about any random variable X, called Chebyshev’s inequality. Suppose X has mean μ and standard deviation σ. Then ... for any value of k. How much of the probability must lie within three standard deviations of the mean?
Get solution
6.9.28 There is a general inequality about any random variable X, called Chebyshev’s inequality. Suppose X has mean μ and standard deviation σ. Then ... for any value of k. Compare
the result of Exercise 26 with the second rule of thumb for bell-shaped
distributions. Which gives more precise information?
Get solution
6.9.19 Find
the probability that the random variable has a value less than one
standard deviation below the expectation and less than two standard
deviations below the expectation. How do the results compare with the
rules of thumb for a bell-shaped distribution? The p.d.
f. of a random variable X is f (x)=2x for 0 ≤ x ≤1 (as in Exercise 13).
Get solution
6.9.30 Consider again the salaries presented in Section 6.8, Exercises 27 and 28. ... For
each, find MAD, the variance, and coefficient of variation. Which
statistics are most sensitive to large values? What are the units of
each statistic? With the top salary of $4,500,000.
Get solution
6.9.31
For
the data first presented in Section 6.6, Exercises 37–40, find the
variance and standard deviation. How many of the values lie within two
standard deviations of the mean? ... Experiment a (Section 6.7,
Exercise 23 computes the expectation). Exercise 23 For
the data presented in Section 6.6, Exercises 37–40, write the results
in terms of a random variable and find the expectation. What fraction of
experiments have a result less than the expectation? Experiment a
Get solution
6.9.32
For
the data first presented in Section 6.6, Exercises 37–40, find the
variance and standard deviation. How many of the values lie within two
standard deviations of the mean? ... Experiment b (Section 6.7,
Exercise 24 computes the expectation). Exercise 24 For
the data presented in Section 6.6, Exercises 37–40, write the results
in terms of a random variable and find the expectation. What fraction of
experiments have a result less than the expectation? Experiment b
Get solution
6.9.33 For
the data first presented in Section 6.6, Exercises 37–40, find the
variance and standard deviation. How many of the values lie within two
standard deviations of the mean? ... Experiment c (Section 6.7, Exercise 25 computes the expectation).Exercise 25 For
the data presented in Section 6.6, Exercises 37–40, write the results
in terms of a random variable and find the expectation. What fraction of
experiments have a result less than the expectation? Experiment c
Get solution
6.9.34
For
the data first presented in Section 6.6, Exercises 37–40, find the
variance and standard deviation. How many of the values lie within two
standard deviations of the mean? ... Experiment d (Section 6.7,
Exercise 26 computes the expectation). Exercise 26 For
the data presented in Section 6.6, Exercises 37–40, write the results
in terms of a random variable and find the expectation. What fraction of
experiments have a result less than the expectation? Experiment d
Get solution
6.9.35
Estimate the standard deviation, coefficient of variation, 2.5th
percentile, and 97.5th percentile from the following figures. ...
Get solution
6.9.26 There is a general inequality about any random variable X, called Chebyshev’s inequality. Suppose X has mean μ and standard deviation σ. Then ... for any value of k. What does this mean for k =2? How much of the probability must lie within two standard deviations of the mean?
Get solution
6.9.37
Estimate the standard deviation, coefficient of variation, 2.5th
percentile, and 97.5th percentile from the following figures. ...
Get solution
6.9.38
Estimate the standard deviation, coefficient of variation, 2.5th
percentile, and 97.5th percentile from the following figures. ...
Get solution
6.9.39 Draw bell-shaped p.d.f.’s with the following properties. A p.d.
f. with the same standard deviation of 10, but with a mean of 500. Calculate the coefficient of variation.
Get solution
6.9.40 Draw bell-shaped p.d.f.’s with the following properties. A p.d.
f. with the same standard deviation of 10, but with a mean of 5. Calculate the coefficient of variation.
Get solution
6.9.41 Draw bell-shaped p.d.f.’s with the following properties. A p.d.
f. with mean of 50 and coefficient of variation of 0.4. Calculate the standard deviation.
Get solution
6.9.42 Draw bell-shaped p.d.f.’s with the following properties. A p.d.
f. with mean of 50 and coefficient of variation of 0.1. Calculate the standard deviation.
Get solution
6.9.43 Suppose a population obeys ... where ... is a random variable that takes on the value 1.5 with probability 0.6 and 0.5 with probability 0.4. Suppose ... =1. Find the variance of the random variable ....
Get solution
6.9.44 Suppose a population obeys ... Find the variance of the random variable ln(...).
Get solution
6.9.45 The bell-shaped or normal p.d.
f. has the equation ... μ is the mean and σ the standard deviation. Set μ=50 and σ =20.
a. Plot this function between some reasonable limits.
b. Use integration to compute the mean and standard deviation. Remember that the limits of integration are from −∞ to ∞ .
c. Find the coefficient of variation. d.
We can check the rules of thumb regarding the shape of the curve and
the standard deviation. Find the first derivative and solve for the
maximum. Does it match the mean?
e. Find the second derivative and
solve for the points of inflection. Do they match the rules of thumb?
46. Find and graph the cumulative distribution F associated with
Get solution
6.9.46 Find and graph the cumulative distribution F associated with f from
the previous problem. Compute the percentiles associated with points
one and two standard deviations above and below the mean. Mark these on
your graph. How well do they match the rules of thumb? Find the 5th and
95th percentile and the lower and upper quartiles.
Get solution
