Continuous rv analysis: probability, graphs and density functions (chapter 8) môn Xác suất thống kê| Trường Đại học Ngoại Thương

8.4 A random variable is uniformly distributed between 100 and 150. a. Draw the density function.
b. Find P(X>110).
8.15 Here is another density function.
c. Find P(120 <X<135).
f1x2=.40 0 <x<1
d. Find P(X<122). =.05 1 <x<13
8.12 The following function is the density function for a. Graph the density function. the rfa(n x d ) o = m x− v 1 a 8 r i a b 1 l x X < : 5
b. Determine the probability that X is less than 8.
c. What is the probability that X lies between .4 and10? a. Graph the density function.
8.16 The following density function describes the ran-
b. Find the probability that X lies between 2 and 4. dom variable X.
c. What is the probability that X is less than 3?
f1x2=.10 0 <x<2 f(x)=dx
8.13 The following density function describes the ran- = .20 2 < x < 5 dom variable X. 25 0 = .15 5 < x < 6 <x<5 = .05 6 < x < 7 25 5<x<10 a. Graph the density function.
b. Calculate the probability that X is less than 5.5. 10 −x
c. Calculate the probability that X is greater than 3.5.
d. What is the probability that X lies between 1 a. Graph the density function. and6.5?
b. Find the probability that X lies between 1 and 3.
c. What is the probability that X lies between 4 and 8?
8.17 Here is another function.
d. Compute the probability that X is less than 7.
f(x) = .2x 0 < x < 2
e. Find the probability that X is greater than 3. = .4 2 < x < 3.5
8.14 The following is a graph of a density function.
a. Confirm that it is a density function. b. Graph the function.
c. Determine the probability that X is less than 2. f(x)
d. Find the probability that X is less than 3.
e. What is the probability that X lies between 1 and2.5? .10
8.18 The following density function describes the ran- dom variable X.
f(x) = .40 − .10x 0 < x < 4
= .10x − .40 4 < x < 6 0 x a. Graph the density function. 20
a. Determine the density function.
b. What is the probability that X is less than 2?
b. Find the probability that X is greater than 10.
c. Find the probability that X is greater than 5.
c. Find the probability that X lies between 6 and 12.
d. Find the probability that X lies between 2.5
CONTINUOUS PROBABILITY DISTRIBUTIONS 265
a student loan is selected at random. Find the fol- lowing probabilities.
a. The loan is greater than $30,000.
b. The loan is less than $22,500.
c. The loan falls between $20,000 and $32,000.
8.48 The Tesla Model S 85D is an electric car that the
manufacturer claims can travel 270 miles on a single
charge. However, the actual distance depends on a
number of factors including speed and whether the
car is driven in the city or on highways. Suppose
that the distance is a normally distributed random
variable with a mean of 200 miles and a standard
deviation of 20 miles. An owner of this model
intends to travel to a nearby city and return on the
same charge. If the total distance is 210 miles, what
is the probability that car makes it without running out of power?
8.39 Xis normally distributed with mean 250 and stan-
dard deviation 40. What value of X does only the top 15% exceed?
8.40 Xis normally distributed with mean 1,000 and
standard deviation 250. What is the probability that
X lies between 800 and 1, 100?
8.50 Economists frequently make use of quintiles (i.e.,
8.41 Xis normally distributed with mean 50 and stan-
the 20th, 40th, 60th, and 80th percentiles) particu-
dard deviation 8. What value of X is such that only
larly when discussing incomes. Suppose that in a 8% of values are below it?
large city household incomes are normally distrib-
uted with a mean of $50,000 and a standard devia-
8.42 The long-distance calls made by the employees of
tion of $10,000. An economist wishes to identify the
a company are normally distributed with a mean of
quintiles. Unfortunately, he did not pass his statis-
6.3 minutes and a standard deviation of 2.2 minutes.
tics course. Help him by providing the quintiles.
Find the probability that a call
a. lasts between 5 and 10 minutes.
8.51 The top-selling Red and Voss tire is rated 70,000 b. lasts more than 7 minutes.
miles, which means nothing. In fact, the distance c. lasts less than 4 minutes.
the tires can run until they wear out is a normally
distributed random variable with a mean of 82,000
miles and a standard deviation of 6,400 miles.
a. What is the probability that a tire wears out before 70,000 miles?
8.44 The lifetimes of lightbulbs that are advertised to
b. What is the probability that a tire lasts more
last for 5,000 hours are normally distributed with than 100,000 miles?
a mean of 5,100 hours and a standard deviation of
200 hours. What is the probability that a bulb lasts
8.52 The heights of 2-year-old children are normally dis-
longer than the advertised figure?
tributed with a mean of 32 inches and a standard devi-
ation of 1.5 inches. Pediatricians regularly measure
the heights of toddlers to determine whether there is
a problem. There may be a problem when a child is
in the top or bottom 5% of heights. Determine the
8.46 SAT scores are normally distributed with a mean
heights of 2-year-old children that could be a problem.
of 1,000 and a standard deviation of 300. Find the quartiles.
8.47 According to a PEW Research Center survey,
the mean student loan at graduation is $25,000.
Suppose that student loans are normally distributed
with a standard deviation of $5,000. A graduate with
observed that its batteries last for an average of
26hours when used in a toy racing car. The amount
of time is normally distributed with a standard devi- ation of 2.5 hours.
a. What is the probability that the battery lasts between 24 and 28 hours?
b. What is the probability that the battery lasts lon- ger than 28 hours?
8.56 The amount of time devoted to studying statistics each
c. What is the probability that the battery lasts less
week by students who achieve a grade of A in the course than 24 hours?
is a normally distributed random variable with a mean
of 7.5 hours and a standard deviation of 2.1 hours.
8.62 Because of the relatively high interest rates, most
a. What proportion of A students study for more
consumers attempt to pay off their credit card bills than 10 hours per week?
promptly. However, this is not always possible. An
b. Find the probability that an A student spends
analysis of the amount of interest paid monthly by
between 7 and 9 hours studying.
a bank’s Visa cardholders reveals that the amount
c. What proportion of A students spend fewer than
is normally distributed with a mean of $27 and a 3 hours studying? standard deviation of $7.
d. What is the amount of time below which only
a. What proportion of the bank’s Visa cardholders
5% of all A students spend studying? pay more than $30 in interest?
b. What proportion of the bank’s Visa cardholders
8.57 The number of pages printed before replacing the pay more than $40 in interest?
cartridge in a laser printer is normal y distributed
c. What proportion of the bank’s Visa cardholders
with a mean of 11,500 pages and a standard deviation pay less than $15 in interest?
of 800 pages. A new cartridge has just been installed.
d. What interest payment is exceeded by only 20%
a. What is the probability that the printer produces
of the bank’s Visa cardholders?
more than 12,000 pages before this cartridge must be replaced?
8.63 It is said that sufferers of a cold virus experience
b. What is the probability that the printer produces
symptoms for 7 days. However, the amount of time fewer than 10,000 pages?
is actually a normally distributed random variable
whose mean is 7.5 days and whose standard devia- tion is 1.2 days.
a. What proportion of cold sufferers experience fewer than 4 days of symptoms?
b. What proportion of cold sufferers experience
symptoms for between 7 and 10 days?
8.59 The mean monthly income of graduates of pro-
8.64 How much money does a typical family of four
fessional and Ph.D. degrees is $6,000 according
spend at a McDonald’s restaurant per visit? The
to a recent PEW Research Center survey. If these
amount is a normally distributed random variable
incomes are normally distributed with a standard
with a mean of $16.40 and a standard deviation of deviation of $1,200, $2.75.
a. What proportion of incomes is greater than
a. Find the probability that a family of four spends $4,900? less than $10.
b. Calculate the proportion of incomes that fall
b. What is the amount below which only 10% of between $3,800 and $5,700.
families of four spend at McDonald’s?
c. Calculate the proportion of incomes that are less
8.65 The final marks in a statistics course are normally than $6,500.
distributed with a mean of 70 and a standard devia-
8.60 A golfer playing a new course encounters a hole that
tion of 10. The professor must convert all marks to
requires a drive of 145 yards to successful y clear a
letter grades. She decides that she wants 10% A’s,
pond. She knows that her drives are normally distrib-
30%B’s, 40%C’s, 15%D’s, and 5%F’s. Determine
uted with a mean of 155 yards and a standard devia-
the cutoffs for each letter grade.
tion of 9 yards. What is the probability that after her
drive her golf ball wil be at the bottom of the pond?
8.61 Battery manufacturers compete on the basis of
the amount of time their products last in cameras
and toys. A manufacturer of alkaline batteries has