6.7.1 For
the data presented in Section 6.6, Exercises 1–4, 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.Exercises 1–4 Draw
histograms describing the probabilities of the outcomes of four
experiments to count the number of mutants in a bacterial cultural. ...
Get solution
6.7.2 For
the data presented in Section 6.6, Exercises 1–4, 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. Exercises 1–4 Draw
histograms describing the probabilities of the outcomes of four
experiments to count the number of mutants in a bacterial cultural. ...
Get solution
6.7.3 For
the data presented in Section 6.6, Exercises 1–4, 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. Exercises 1–4 Draw
histograms describing the probabilities of the outcomes of four
experiments to count the number of mutants in a bacterial cultural. ...
Get solution
6.7.4 For
the data presented in Section 6.6, Exercises 1–4, 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
6.7.5
Find the expected number of molecules in the cell using the data in
Example 6.7.19 at the following times. At time 4. Reference Example
6.7.19 ...
Get solution
6.7.6
Find the expected number of molecules in the cell using the data in
Example 6.7.19 at the following times. At time 8. Reference Example
6.7.19 ...
Get solution
6.6.7
Find and sketch the cumulative distribution associated with the
histogram from the earlier problem. The histogram in Exercise 3.
Reference Exercise 3. Draw
histograms describing the probabilities of the outcomes of four
experiments to count the number of mutants in a bacterial cultural.
... Experiment c.
Get solution
6.6.8
Find and sketch the cumulative distribution associated with the
histogram from the earlier problem. The histogram in Exercise 4.
Reference Exercise 4. Draw
histograms describing the probabilities of the outcomes of four
experiments to count the number of mutants in a bacterial cultural.
... Experiment d.
Get solution
6.6.9 On each histogram, find the most and least likely simple events. Is the histogram symmetric? ...
Get solution
6.6.10 On each histogram, find the most and least likely simple events. Is the histogram symmetric? ...
Get solution
6.7.11 For
each continuous random variables with the given p.d.f., find a discrete
random variable that approximates it. Graph the histogram for this
discrete random variable and compare it with the p.d.
f. for the
continuous random variable. Find the expectation of the discrete random
variable, and check whether it is equal to the expectation of the
continuous random variable. 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 7). Suppose that measurements are very imprecise, and that all values of X ≤0.5 are recorded as 0.25 and all values of X >0.5
are recorded as 0.75. Write a random variable describing these
imprecise measurements, find the associated probabilities, and compute
the expectation.
Get solution
6.6.12 On each histogram, find the most and least likely simple events. Is the histogram symmetric? ...
Get solution
6.7.13 For
each continuous random variables with the given p.d.f., find a discrete
random variable that approximates it. Graph the histogram for this
discrete random variable and compare it with the p.d.
f. for the
continuous random variable. Find the expectation of the discrete random
variable, and check whether it is equal to the expectation of the
continuous random variable. 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 7). Suppose that measurements are imprecise, and that all values of X ≤0.25 are recorded as 0.25, all values of 0.25 < X ≤0.5 are recorded as 0.5, all values of 0.5 < X ≤0.75 are recorded as 0.75, and all values of 0.75 < X are recorded as 1.0.
Get solution
6.7.14 For
each continuous random variables with the given p.d.f., find a discrete
random variable that approximates it. Graph the histogram for this
discrete random variable and compare it with the p.d.
f. for the
continuous random variable. Find the expectation of the discrete random
variable, and check whether it is equal to the expectation of the
continuous random variable. The p.d.
f. of a random variable X is ...for 0 ≤ x ≤2 (as in Section 6.6, Exercises 18 and 8). Suppose that measurements are imprecise, and that all values of X ≤0.5 are recorded as 0.25, all values of 0.5 < X ≤1.0 are recorded as 0.75, all values of 1.0 < X ≤1.5 are recorded as 1.25, and all values of 1.5 < X are recorded as 1.75.
Get solution
6.7.15 Check that the following could be p.d.f.’s and compute their expectations. Does anything seem odd about them? ...
Get solution
6.7.16 Check that the following could be p.d.f.’s and compute their expectations. Does anything seem odd about them? ...
Get solution
6.7.17 Think
about one or more molecules independently leaving a cell, each with
probability 0.9 in a given second. Find the random variable describing
the following events, find the probabilities of the outcomes, and find
the expectation. A Bernoulli random variable describing whether a molecule is in or out at time 1.
Get solution
6.7.18 Think
about one or more molecules independently leaving a cell, each with
probability 0.9 in a given second. Find the random variable describing
the following events, find the probabilities of the outcomes, and find
the expectation. A Bernoulli random variable describing whether a molecule is in or out at time 2.
Get solution
6.7.19
Think
about one or more molecules independently leaving a cell, each with
probability 0.9 in a given second. Find the random variable describing
the following events, find the probabilities of the outcomes, and find
the expectation. A Bernoulli random variable describing whether two
molecules are together (both in or both out) or separate at time 1.
Get solution
6.7.20 Think
about one or more molecules independently leaving a cell, each with
probability 0.9 in a given second. Find the random variable describing
the following events, find the probabilities of the outcomes, and find
the expectation. A Bernoulli random variable describing whether three out of three molecules remain inside at time 1.
Get solution
6.7.22 Think
about one or more molecules independently leaving a cell, each with
probability 0.9 in a given second. Find the random variable describing
the following events, find the probabilities of the outcomes, and find
the expectation. A discrete random variable that counts the number out of two molecules that are in at time 2.
Get solution
6.6.22 Find and sketch the c.d.
f. associated with the given p.d.
f. and check that it increases to a value of 1. ...
Get solution
6.7.23 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
6.6.24 Find and sketch the c.d.
f. associated with the given p.d.
f. and check that it increases to a value of 1. The p.d.
f. is g(t)=6t (1 − t) for 0 ≤ t ≤1 (as in Exercise 20).
Get solution
6.7.25 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
6.7.26 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
6.6.27 Find the probability in two ways:
a. By integrating the given p.d.
f.
b. By using the c.d.
f. Make sure that your answers match. Shade the given areas on a graph of the p.d.
f. The p.d.
f. is ...for 1≤t ≤e (as in Exercises 19 and 23). Find the probability that the measurement is between 2.0 and 2.5.
Get solution
6.7.28 Consider
again the cells in Section 6.4, Exercises 29 and 30, but suppose that
older cells, instead of not staining as often, do not stain as well. In
particular, they produce a brightness of 7, while young cells have a
brightness of 9. Write a random variable describing the brightness, and
find its probability distribution and its expectation. Suppose 70% of the cells are young.
Get solution
6.6.29 Sketch the c.d.
f. associated with each of the following p.d.f.’s. ...
Get solution
6.6.30 Sketch the c.d.
f. associated with each of the following p.d.f.’s. ...
Get solution
6.6.31 Sketch the p.d.
f. associated with each of the following c.d.f.’s. ...
Get solution
6.7.32
Suppose immigration and emigration change the sizes of four
populations with the following probabilities. ... Write the
result as a random variable and find the expectation. Population
b. How many immigrants do you think would arrive (or depart) in 10 years? Will the population grow?
Get solution
6.7.33
Suppose immigration and emigration change the sizes of four
populations with the following probabilities. ... Write the
result as a random variable and find the expectation. Population
c. How many immigrants would arrive (or depart) in 10 years? Will the
population grow? Does the expectation seem close to the “middle” of the
distribution?
Get solution
6.7.34
Suppose immigration and emigration change the sizes of four
populations with the following probabilities. ... Write the
result as a random variable and find the expectation. Population
d. About how many immigrants would arrive (or depart) in 10 years? Will
the population grow? Does the expectation seem close to the “middle” of
the distribution?
Get solution
6.6.35 Draw
histograms of the distributions of cell age from the assumptions in the
earlier problem. Find and graph the cumulative distribution. The cells in Section 6.4, Exercise 31, where Pr(cell is 0 day old) = 0.4 Pr(cell is 1 day old) = 0.3 Pr(cell is 2 days old) = 0.2 Pr(cell is 3 days old) = 0.1.
Get solution
6.7.36 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. Use the following indefinite integral fact to find the expectation for the following values of λ. ... λ=2.0.
Get solution
6.6.37
One
hundred pairs of plants are crossed, and each pair produces ten
offspring. The number of tall offspring is then counted. For the given
experiment, draw a histogram of the probability of each result, and find
the requested probability. ... Draw the histogram for
experiment a, and find the probability that between 4 and 6 plants
(inclusive) are tall.
Get solution
6.7.38
The formula for the expectation in mathematics is identical to the
formula for the center of mass in
physics. Mass acts like probability, and distance acts like the
measured random variable. For example, suppose two people are on a see
saw, and one is 2 m to the right of center and weighs 60 kg, and the
other is 3 m to the right of center and weighs 50 kg. The proportion of
mass in the first person is 60/110, and the proportion in the second is
50/110. The center of mass is .... In terms of balancing, these two
people act like a single 110 kg person at a position 2.455 meters to the
right of center. A
20 kg child is 3 m to the right of center, her 30 kg older brother is 1
m to the right of center, and an 80 kg adult is 1 m to the left of
center. Find the center of mass. Who will go up?
Get solution
6.6.39
One
hundred pairs of plants are crossed, and each pair produces ten
offspring. The number of tall offspring is then counted. For the given
experiment, draw a histogram of the probability of each result, and find
the requested probability. ... Draw the histogram for
experiment c, and find the probability that between 4 and 6 plants
(inclusive) are tall.
Get solution
6.7.40
The formula for the expectation in mathematics is identical to the
formula for the center of mass in
physics. Mass acts like probability, and distance acts like the
measured random variable. For example, suppose two people are on a see
saw, and one is 2 m to the right of center and weighs 60 kg, and the
other is 3 m to the right of center and weighs 50 kg. The proportion of
mass in the first person is 60/110, and the proportion in the second is
50/110. The center of mass is .... In terms of balancing, these two
people act like a single 110 kg person at a position 2.455 meters to the
right of center. A
20 kg child is 3 m to the right of center, her 30 kg older brother is 1
m to the right of center, and an 80-kg adult wishes to balance the see
saw by sitting a distance x to the left of center. Solve for x.
Get solution
6.7.42 The
relation between the mathematical expectation and the center of mass in
physics also holds for continuous distributions. Mass density acts
like probability density (after the mass density has been divided by the
total density). For example, suppose the density of a 1 m long bar is ρ(x)=4x kg/m. Then the total mass is .... Dividing ρ(x) by this total mass of 2 gives a density f (x)=2x, which has integral 1, like a p.d.
f. The xpectation of a random variable X with this p.d.
f. is ... Find the center of mass of a bar with mass density ρ(x)= ... (2 − x) for 0≤ x ≤2. Is it to the left or the right of the center of the bar at x =1? Is it at the point where ρ(x) takes on its maximum?
Get solution
6.6.42 An
experiment to see which color of male birds female birds prefer is
repeated two times. The first time, females mate with red males with
probability 0.5, with blue males with probability 0.3, and with green
males with probability 0.2. The second time, females mate with red males
with probability 0.4, with blue males with probability 0.35, and with
green males with probability 0.25. At the end, the results of the two
experiments are combined. Suppose that 100 female birds were
tested in the first experiment and 200 females in the second. Find the
number out of 300 that mated with each type of male, and convert the
results into a probability distribution.
Get solution
6.7.43 The
relation between the mathematical expectation and the center of mass in
physics also holds for continuous distributions. Mass density acts
like probability density (after the mass density has been divided by the
total density). For example, suppose the density of a 1 m long bar is ρ(x)=4x kg/m. Then the total mass is .... Dividing ρ(x) by this total mass of 2 gives a density f (x)=2x, which has integral 1, like a p.d.
f. The xpectation of a random variable X with this p.d.
f. is ... As
in Exercise 37, a 4 m long wooden board has a fulcrum placed 1 m from
the left end. A 20 kg child sits at the right end of the board (at
position x =3) and an 80 kg adult sits at the left end (at position x =−1). Suppose that the board has density of 5 kg/m. Find the center of mass. Who will go up
Get solution
6.7.44 The
relation between the mathematical expectation and the center of mass in
physics also holds for continuous distributions. Mass density acts
like probability density (after the mass density has been divided by the
total density). For example, suppose the density of a 1 m long bar is ρ(x)=4x kg/m. Then the total mass is .... Dividing ρ(x) by this total mass of 2 gives a density f (x)=2x, which has integral 1, like a p.d.
f. The xpectation of a random variable X with this p.d.
f. is ... As
in Exercise 37, a 4 m long wooden board has a fulcrum placed 1 m from
the left end. A 20 kg child sits at the right end of the board (at
position x =3), and an 80 kg adult sits at the left end (at position x =−1). Suppose that the board has density of y kg/m. Find the value of y for which the board will balance.
Get solution
6.7.45 Your
computer should have a way to roll a random die (giving results 1
through 6 each with equal probability). Roll such a die 5, 10, 20, 50,
and 100 times and find the average score. How close is each to the
expectation?
Get solution
6.6.46 The p.d.
f. for the waiting time X until an event occurs often follows the exponential distribution (to be studied in Section 7.6), with the form ...for some positive value of α, defined for x ≥0. For each of the following values of α,
a. Find the c.d.
f.
b. Plot the p.d.
f. and c.d.
f.
c. Check that the p.d.
f. is the derivative of the c.d.
f.
d. Find Pr(X ≤1). Indicate this on both of your graphs.
e. Find Pr(1≤ X ≤3).
f. Find Pr(1 ≤ X ≤1.01) and show that it is approximately g (1) · 0.01. α =2.0.
Get solution
6.7.47 Your
computer should have a way to choose a random number between 0 and 1
(using the uniform p.d.f.). Pick 5, 10, 20, 50, and 100 such numbers and
find the average. How close is each to the expectation?
Get solution
