7.6.1 Compute the following probabilities. In each case, sketch
the probability distribution and shade the associated area. The
probability that the first success is on the third trial if each trial
has a probability 0.2 of success.
Get solution
7.6.2
Compute the following probabilities. In each case, sketch the
probability distribution and shade the associated area. The probability
that the first success is on the fifth trial if each trial has a
probability 0.3 of success.
Get solution
7.6.3
Compute the following probabilities. In each case, sketch the
probability distribution and shade the associated area. The probability
that the first success occurs on or before the fourth trial if each has
a probability 0.2 of success.
Get solution
7.6.4
Compute the following probabilities. In each case, sketch the
probability distribution and shade the associated area. The probability
that the first success occurs on or before the third trial if each has a
probability 0.3 of success.
Get solution
7.6.5
Compute the following probabilities. In each case, sketch the
probability distribution and shade the associated area. The probability
that the first success occurs on or after the third trial if each has a
probability 0.2 of success.
Get solution
7.6.6
Compute the following probabilities. In each case, sketch the
probability distribution and shade the associated area. The probability
that the first success occurs on or after the sixth trial if each trial
has a probability 0.3 of success.
Get solution
7.6.7
Compute
the following probabilities. In each case, sketch the probability
distribution function and shade the area associated with the question.
Events occur at a rate of 0.5/s. Find the probability that the first
event occurs between times 1.0 and 2.0.
Get solution
7.6.8
Compute
the following probabilities. In each case, sketch the probability
distribution function and shade the area associated with the question.
Events occur at a rate of 1.5/s. Find the probability that the first
event occurs between times 0.2 and 1.0.
Get solution
7.6.9
Compute
the following probabilities. In each case, sketch the probability
distribution function and shade the area associated with the question.
Events occur at rate 0.2/s. Find the probability that the first event
occurs before t =1 or after t =3.
Get solution
7.6.10
Compute
the following probabilities. In each case, sketch the probability
distribution function and shade the area associated with the question.
Events occur at rate 5.0/s. Find the probability that the first event
occurs before t =0.1 or after t =0.5.
Get solution
7.6.11 Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success. The probability of success is q =0.2.
Get solution
7.6.12 Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success. The probability of success is q =0.3.
Get solution
7.6.13 Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success. The probability of success is q =0.7.
Get solution
7.6.15 Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ. The rate is λ=0.2.
Get solution
7.6.15 Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ. The rate is λ=0.2.
Get solution
7.6.16 Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ. The rate is λ=0.5.
Get solution
7.6.17 Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ. The rate is λ=1.5.
Get solution
7.6.18 Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ. The rate is λ=5.0.
Get solution
7.6.19 We can verify that the cumulative distribution for the geometric distribution is ...with a mathematical trick. Show that ... for any x by multiplying both sides by 1 − x and working out the algebra.
Get solution
7.6.20 We can verify that the cumulative distribution for the geometric distribution is ...with a mathematical trick. ...
Get solution
7.6.21 We found that the function ... is equal to the Taylor series ... when 0 < x <1 (Example 3.7.10). Use this Taylor series to show that ... for any 0<q <1.
Get solution
7.6.22 We found that the function ... is equal to the Taylor series ... when 0 < x <1
(Example 3.7.10). Differentiate the Taylor series term by term and use
it to derive the expectation of a geometric random variable.
Get solution
7.6.23 We
can find the exponential p.d.
f. as the limit of the geometric
distribution. Think of dividing time up into small units of length Δt. If the probability of leaving during 1 min is q, what is the approximate probability of leaving during a short interval of time Δt?
Get solution
7.6.24 We
can find the exponential p.d.
f. as the limit of the geometric
distribution. Think of dividing time up into small units of length Δt. Find the number of steps of duration Δt during an interval of length t.
Get solution
7.6.25 We
can find the exponential p.d.
f. as the limit of the geometric
distribution. Think of dividing time up into small units of length Δt. Use
the geometric distribution with the probability in Exercise 23 and the
number of steps in Exercise 24 to find the approximate probability that a
molecule is still inside at time t.
Get solution
7.6.26 We
can find the exponential p.d.
f. as the limit of the geometric
distribution. Think of dividing time up into small units of length Δt. Find the limit of the result of Exercise 25, using the definition of the number e (Definition 2.8 in Subsection 2.8.1). What exponential distribution has this expression for its survivorship function?
Get solution
7.6.27
The expectation of the geometric distribution can be found using a
clever trick. A molecule either leaves immediately or not. Find the
probability that it leaves immediately and the time at which it leaves.
Get solution
7.6.28
The expectation of the geometric distribution can be found using a
clever trick. Find the probability that a molecule does not leave
immediately. Show that the expected time to leave in this case is 1 + E(T ).
Get solution
7.6.29
The expectation of the geometric distribution can be found using a
clever trick. Multiply the values by the probabilities in Exercises 27
and 28 to find an expression for E(T ).
Get solution
7.6.30 The expectation of the geometric distribution can be found using a clever trick. Solve for E(T ).
Get solution
7.6.31 The
variance of the geometric distribution can be found with a clever
trick, much like that in Exercises 27–30. Define the random variable R, which is 1 with probability q and 1 + T with probability 1 − q. Why does R have the same probability distribution as T ?
Get solution
7.6.32 The
variance of the geometric distribution can be found with a clever
trick, much like that in Exercises 27–30. Define the random variable R, which is 1 with probability q and 1 + T with probability 1 − q. Show that ...
Get solution
7.6.33 The
variance of the geometric distribution can be found with a clever
trick, much like that in Exercises 27–30. Define the random variable R, which is 1 with probability q and 1 + T with probability 1 − q. Why is ...
Get solution
7.6.34 The
variance of the geometric distribution can be found with a clever
trick, much like that in Exercises 27–30. Define the random variable R, which is 1 with probability q and 1 + T with probability 1 − q. Solve for ...and use the computational formula to find Var(T ).
Get solution
7.6.35 Use integration by parts to compute the following statistics for an exponentially distributed random variable with parameter λ. The expectation.
Get solution
7.6.36 Use integration by parts to compute the following statistics for an exponentially distributed random variable with parameter λ. The variance
Get solution
7.6.37 Find the probabilities of the following events. A
certain highly mutable gene has a 1.0% chance of mutating each time a
cell divides (as in Section 6.2, Exercise 25). Suppose that there are 15
cell divisions between each pair of generations. What is the chance
that the gene first mutates during the last division? What is the chance
that the gene mutates at some point during the first 15 divisions?
Get solution
7.6.38 Find the probabilities of the following events. A
herd of lemmings is standing at the top of a cliff, and each jumps off
with probability 0.2 each hour (as in Section 6.2, Exercise 24). What is
the probability that a lemming first jumps on the fifth hour? What is
the probability that a particular lemming has not jumped by the end of
third hour?
Get solution
7.6.39 Find the probabilities of the following events. A
molecule has a 5.0% chance of binding to an enzyme each second and
remains permanently attached thereafter (as in Section 6.2, Exercise
25). Find the probability that it binds during the tenth second, and the
probability that it binds n the tenth second or before
Get solution
7.6.40 Find the probabilities of the following events. In
tropical regions, growing caterpillars can be eaten with probability
0.15 each day (as in Section 6.2, Exercise 26). What is the probability
that a caterpillar is eaten on the fourth day? If a caterpillar takes 25
days to develop, what is the probability it survives?
Get solution
7.6.41 Find the probabilities of the following events. A
light bulb blows out with probability 0.01 each day. What is the
probability that it blows out on the 50th day? What is the probability
it blows out after more than 200 days?
Get solution
7.6.42 Find the probabilities of the following events. 10%
of some type of item are defective. Find the probability that the first
defective item found is the fifth one inspected. What is the
probability that a defective one is found if five are inspected?
Get solution
7.6.43 lab
is screening to find mutant flies. 10% of the offspring of mutant flies
have purple eyes, while none of the offspring of wild type flies have
purple eyes. Suppose that purple-eyed flies are just as easy to catch as
ordinary flies. Use the binomial distribution to find the expected number of purple-eyed flies (from a mutant parent) after n have been checked. How many must be checked for the expectation to equal 1?
Get solution
7.6.44 lab
is screening to find mutant flies. 10% of the offspring of mutant flies
have purple eyes, while none of the offspring of wild type flies have
purple eyes. Suppose that purple-eyed flies are just as easy to catch as
ordinary flies. What is the expected number of mutant flies
that must be inspected to find the first one with purple eyes? What is
the expected number of purple-eyed flies that will be found if this many
are checked? What is the probability that exactly one purple-eyed fly
is found if this many are checked?
Get solution
7.6.45 lab
is screening to find mutant flies. 10% of the offspring of mutant flies
have purple eyes, while none of the offspring of wild type flies have
purple eyes. Suppose that purple-eyed flies are just as easy to catch as
ordinary flies. Suppose that half of the parent flies are known
to be mutants. Of 20 offspring flies inspected, none have purple eyes.
What is the conditional probability that the parents are mutants? If
parents are discarded if all 20 offspring have normal eyes, what
fraction of mutant parents will be discarded?
Get solution
7.6.46 lab
is screening to find mutant flies. 10% of the offspring of mutant flies
have purple eyes, while none of the offspring of wild type flies have
purple eyes. Suppose that purple-eyed flies are just as easy to catch as
ordinary flies. Suppose that only 5% of the parent flies are
known to be mutants. Of 20 flies inspected, none are found to be have
purple eyes. What is the conditional probability that the parents are
mutants? If parents are discarded if all 20 offspring have normal eyes,
what fraction of mutant parents will be discarded?
Get solution
7.6.48 Inspecting
for defective items until the first is found only works when all the
items to be checked have the same probability of being defective.
Suppose that items to be inspected had been deceptively sorted in the
factory in order from best to worst. The first item is defective with
probability 0.1, the second with probability 0.2, and so forth. Find the whole probability distribution.
Get solution
7.6.48 Inspecting
for defective items until the first is found only works when all the
items to be checked have the same probability of being defective.
Suppose that items to be inspected had been deceptively sorted in the
factory in order from best to worst. The first item is defective with
probability 0.1, the second with probability 0.2, and so forth. Find the whole probability distribution.
Get solution
7.6.49 Find
the probabilities of the following events. Find the associated p.d.f.,
c.d.f., and survivorship function, and give the expectation and
variance. A molecule leaves a cell at rate λ=0.3/s.
What is the probability it has left by the end of the third second? By
the end of the first millisecond? Compare your last answer with the
definition of rate.
Get solution
7.6.50 Find
the probabilities of the following events. Find the associated p.d.f.,
c.d.f., and survivorship function, and give the expectation and
variance. A light bulb blows out at a rate of λ=0.001/h.
What is the probability that it blows out in less than 500 h? In less
than 1 h? Compare your last answer with the definition of rate.
Get solution
7.6.51 Find
the probabilities of the following events. Find the associated p.d.f.,
c.d.f., and survivorship function, and give the expectation and
variance. Phone calls arrive at a rate of λ=0.2/h.
What is the probability that there are no calls in 10 h? What is the
probability that a call arrives during the 45 s you spend in the
bathroom?
Get solution
7.6.52 Find
the probabilities of the following events. Find the associated p.d.f.,
c.d.f., and survivorship function, and give the expectation and
variance. Raindrops hit a leaf at rate 7.3/min. What is the probability that the first one hits in less than 0.5 min?
Get solution
7.6.53 A population of 100 bacteria are dying independently at rate λ=2.0. Find the probability that a given bacterium is alive at time t.
Get solution
7.6.54 A population of 100 bacteria are dying independently at rate λ=2.0. What distribution describes the population at time t? Find the expectation and variance of the number alive as functions of time.
Get solution
7.6.55
Compute the following probabilities. The gene in Exercise 37 has not
mutated after ten divisions. What is the probability that it mutates by
the 15th division?
Get solution
7.6.56
Compute the following probabilities. The caterpillar in Exercise 40
is still alive after 10 days. What is the probability that it is eaten
by day 25?
Get solution
7.6.57
Compute the following probabilities. The molecule in Exercise 49 is
still in the cell after 10 s. What is the probability that it leaves
before the 13th second?
Get solution
7.6.58
Compute the following probabilities. The light bulb in Exercise 50
is OK after 1000 h. What is the probability that it blows out before
hour 1500?
Get solution
7.6.59 You
are trapped behind an annoyingly slow driver (a.s.d.) in a long
no-passing zone. A second a.s.d. merges in front of the first, leaving
you twice as trapped. To calm yourself, you attempt to guess which
driver will exit first. If the length of time that cars remain
on the freeway is an exponential distribution, what is the probability
that the new a.s.d. exits first? Under what conditions might this occur?
Get solution
7.6.60 You
are trapped behind an annoyingly slow driver (a.s.d.) in a long
no-passing zone. A second a.s.d. merges in front of the first, leaving
you twice as trapped. To calm yourself, you attempt to guess which
driver will exit first. Under what assumptions would you expect
the new a.s.d. to exit first? Under what assumptions would you expect
the new a.s.d. to exit second?
Get solution
7.6.61 Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years. Write a differential equation describing the probability P(t) the animal is still alive.
Get solution
7.6.62 Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years. Solve the equation with separation of variables (Algorithm 5.2).
Get solution
7.6.63 Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years. Graph the survivorship function.
Get solution
7.6.64 Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years. Find the c.d.
f. and p.d.f.
Get solution
7.6.65 Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years. Find the probability it survives to age 1.
Get solution
7.6.66 Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years. Find
the probability it survives to age 2 conditional on surviving to age 1.
Why might this value be different from that in Exercise 65?
Get solution
7.6.68 Use your computer to find the mean and variance of the exponential distribution with parameter λ.
What fraction of measurements are greater than the mean? What fraction
of measurements are greater than the mean plus one standard deviation?
The mean plus two standard deviations? Why do these results seem so
different from those in Subsection 6.9.3?
Get solution
7.6.68 Use your computer to find the mean and variance of the exponential distribution with parameter λ.
What fraction of measurements are greater than the mean? What fraction
of measurements are greater than the mean plus one standard deviation?
The mean plus two standard deviations? Why do these results seem so
different from those in Subsection 6.9.3?
Get solution
7.6.69
There is a generalization of the geometric distribution called,
confusingly enough, the negative binomial distribution. Under the same
assumptions as the geometric distribution, the random variable ... is the number of failures that precede the r th success. The case r =1 is the geometric distribution. In general, if the probability of a success on each trial is q, then ... Set q =0.1.
a. Plot the distribution with r =1. Find the expectation.
b. Plot the distribution with r =2. Find the expectation. Does this result make sense?
c. How can you think of the random variable ... as the sum of two other random variables?
d. Plot the distribution with r =10. Find the expectation. Does this result make sense? Why is the shape of the distribution so different?
Get solution
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