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ENGDATA LECTURE9

Page |1

CONTINUOUS UNIFORM DISTRIBUTION

The continuous uniform distribution is one of the simplest continuous distributions in statistics.
It is characterized by a density function that is flat and the probability is uniform in a closed interval [A,
B]. It is often called the rectangular distribution because the density function forms a rectangle with base
B-A and constant height 1/(B-A).

The density function of the continuous uniform random variable X on the interval [A, B] is

{
1
f ( x ; A , B )= B− A , A ≤ x ≤ B ,
0 , elsewhere .

The mean and variance of the uniform distribution are

2
A+ B (B− A)
μ= ∧σ 2= .
2 12

Examples:
1. Suppose that a large conference room at a certain company can be reserved for no more than 4
hours. Both long and short conferences occur quite often. In fact, it can be assumed that the
length X of a conference has a uniform distribution on the interval [0, 4]. (a) What is the
probability density function? (b)What is the probability that any given conference lasts at least 3
hours?
2. The daily amount of coffee in liters, dispensed by a machine located in an airport lobby is a
random variable X having a continuous uniform distribution with A=7 and B=10. Find the
probability that on a given day the amount of coffee dispensed by this machine will be (a) at
most 8.8 liters; (b) more than 7.4 liters but less than 9.5 liters; (c) at least 8.5 liters.
3. A bus arrives every 10 minutes at a bus stop. It is assumed that the waiting time for a particular
individual is a random variable with a continuous uniform distribution. (a) What is the
probability that the individual waits more than 7 minutes? (b) What is the probability that the
individual waits between 2 and 7 minutes?
4. Suppose X follows a continuous uniform distribution from 1 to 5. Determine the conditional
probability P( X >2.5∨X ≤ 4) .

NORMAL DISTRIBUTION

The normal distribution is the most important probability distribution in statistics.Its graph (bell-
shaped curve) is called the normal curve.The graph of a normal distribution is a bell-shaped curve that
extends indefinitely in both directions.

Ex: physical measurements in meteorological experiments, rainfall studies and measurements of

manufactured parts

 Pierre Laplace (1749-1827)

 Abraham De Moivre (1667-1745)
ENGDATA LECTURE9
Page |2

 Karl Friedrich Gauss (1777-1855)

The density of the normal random variable X, with mean μ and variance σ 2 , is
−1 2
( x−μ )
1 2σ
2
, −∞< x < ∞ ,
n ( x ; μ , σ )= e
√2 π σ
2
The mean and variance of n(x; μ,σ) are μ and σ , respectively. The standard deviation is
σ .

The normal curve (sometimes called standard curve) Normal curves with μ 1< μ 2 and σ1=σ2

Normal curves with μ 1=μ 2 and σ1 < σ2 Normal curves with μ 1< μ 2 and σ1 < σ2

Properties of Normal Curve

 A probability distribution is said to be normal if the mean, median and mode coincides at a
single point.
 The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at
x=μ .
 The curve is symmetric about a vertical axis through the mean μ .
 The curve has its points of inflection at x=μ ± σ ; it is concave downward if
μ−σ < X < μ+ σ and is concave upward otherwise.
ENGDATA LECTURE9
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 The right and left tails of the normal curve are asymptotic with respect to the horizontal axis.
 The total area under the curve is equal to 1 or 100%.

The Empirical Rule

The 68-95-99.7 Rule of the Normal Curve
 About 68% of the area under the curve falls within 1 standard deviation of the mean.
(−1< z <1)
 About 95% of the area under the curve falls within 2 standard deviations of the mean.
(−2< z<2)
 About 99.7% of the area under the curve falls within 3 standard deviations of the mean.
(−3< z< 3)

AREAS UNDER THE NORMAL CURVE

The curve of any continuous probability distribution or density function is constructed so that
the area under the curve bounded by two ordinates x=x 1 and x=x 2 equals the probability that
the random variable X assumes a value between x=x 1 and x=x 2 .

The distribution of a normal random variable with mean 0 and variance 1 is called a standard
normal distribution.

The z-Score
Fortunately, we are able to transform all the observations of any normal random variable x
to a new set of observations of a normal random variable z . The formula is:
x−μ
z=
σ
where: x = score (observation)
μ = mean
σ = standard deviation
z = z-score

Examples:
1. Given a standard normal distribution, find the area under the curve that lies (a) to the right of
z=1.84 and (b) between z=-1.97 and z=0.86.
2. Given a standard normal distribution, find the value of k such that (a) P(Z>k) = 0.3015 and (b)
P(k<Z<-0.18) = 0.4197.
3. Given a random variable X having a normal distribution with μ=50∧σ =10 , find the
probability that X assumes a value between 45 and 62.
4. Given that X has a normal distribution with μ=300∧σ =50 , find the probability that X
assumes a value greater than 362.
5. Given a normal distribution with μ=40∧σ=¿ 6, find the value of x that has (a) 45% of the
area to the left and (b) 14% of the area to the right.
6. Given a standard normal distribution, find the area under the curve that lies (a) to the right of
z=1.25 ; (b) to the left of z=−0.4 ; (c) to the left of z=0.8 ; (d) between z=0.4
and z=1.3 ; (e) between z=−0.3 and z=0.9 ; and (f) outside z=−1.5 to
z=1.5 .
ENGDATA LECTURE9
Page |4

7. Find the value of z if the area under a standard normal curve (a) to the right of z is 0.3622; (b) to
the left of z is 0.1131; (c) between 0 and z, with z>0, is 0.4838; (d) between –z and z, with z>0, is
0.9500.
8. Given a normal distribution with μ=30∧σ =6 , find (a) the normal curve area to the right of
x=17; (b)the normal curve area to the left of x=22; (c) the normal curve area between x=32 and
x=41; (d) the value of x that has 80% of the normal curve are to the left; € the two values of x
that contain the middle 75% of the normal curve area.

APPLICATIONS OF THE NORMAL DISTRIBUTION

Examples:
1. A certain type of storage battery lasts, on average, 3.0 years with a standard deviation of 0.5
year. Assuming that the battery lives are normally distributed, find the probability that a given
battery will last less than 2.3 years.
2. An electrical firm manufactures light bulbs that have a life, before burn-out, that is normally
distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find the
probability that a bulb burns between 778 and 834 hours.
3. In an industrial process the diameter of a ball bearing is an important component part. The
buyer sets specifications on the diameter to be 3.0 ± 0.01 cm. The implication is that no part
falling outside these specifications will be accepted. It is known that in the process the diameter
of a ball bearing has a normal distribution with mean μ = 3.0 and standard deviation σ = 0.005.
On the average, how many manufactured ball bearings will be scrapped?
4. Gauges are used to reject all components for which a certain dimension is not within
specification 1. 50 ± d . It is known that this measurement is normally distributed with mean
1.50 and standard deviation 0.2. Determine the value d such that the specifications “cover” 95%
of the measurements.
5. The average grade for an exam is 74, and the standard deviation is 7. If 12% of the class is given
As, and the grades are curved to follow a normal distribution, what is the lowest possible A and
the highest possible B? Also, find the sixth decile.
6. A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard
deviation of 2 ohms. Assuming the resistance follows a normal distribution and can be measured
to any degree of accuracy, what percentage of resistors will have a resistance exceeding 43
ohms? Find the percentage of resistances exceeding 43 ohms if resistance is measured to the
nearest ohm.
7. The IQs of 600 applicants of a certain college are approximately normally distributed with a
mean of 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how
many of these students will be rejected on this basis regardless of their other qualifications?
8. A radar unit is used to measure speeds of cars on a motorway. The speeds are normally
distributed with a mean of 90 km/hr. and a standard deviation of 10 km/hr. What is the
probability that a car picked at random is travelling at more than 100 km/hr.?
9. For a certain type of computers, the length of time between charges of the battery is normally
distributed with a mean of 50 hours and a standard deviation of 15 hours. John owns one of
these computers and wants to know the probability that the length of time will be between 50
and 70 hours.
10. A large group of students took a test in Physics and the final grades have a mean of 70 and a
standard deviation of 10. If we can approximate the distribution of these grades by a normal
distribution, what percent of the students (a) scored higher than 80? (b) should pass the test
ENGDATA LECTURE9
Page |5

11. The SAT is a test for readiness of students for college. The average SAT score (on a 1600 point
scale) is 1025 points and the standard deviation is 200 points. How well must Jessica do on the
SAT in order to place in the top 10% of all students?
12. An intelligence quotient, or IQ, is a score derived from one of several standardized tests
designed to assess intelligence. The mean score is normalized as 100 and the standard deviation
is roughly 15. An IQ of 150 is required for entrance into a gifted program, what percentage of
students is considered eligible?
13. At the New Age Information Corporation, the ages of all new employees hired during the last 5
years are normally distributed. Within this curve, 95.4% of the ages, centered about the mean,
are between 24.6 and 37.4 years. Find the mean age and the standard deviation of the data.
14. The heights of 1000 students are normally distributed with a mean of 174.5 centimeters and a
standard deviation of 6.9 centimeters. Assuming that the heights are recorded to the nearest
half-centimeter, how many of these students you would expect to have heights (a) less than
160.0 centimeters? (b) between 171.5 and 182.0 centimeters inclusive? (c) equal to 175.0
centimeters? (d) greater than or equal to 188.0 centimeters?
15. The weights of a large number of miniature poodles are approximately normally distributed with
a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements are recorded
to the nearest tenth of a kilogram, find the fraction of these poodles with eights (a) over 9.5
kilograms; (b) of at most 8.6 kilograms; (c) between 7.3 and 9.1 kilograms inclusive.

NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

2
If X is a binomial random variable with mean μ=np and variance σ =npq , then the
limiting form of the distribution of
X−np
Z= As n → ∞ , is the standard normal distribution n( z ; 0,1) .
√ npq
Examples:
1. The probability that a patient recovers from a rare blood disease is 0.4. If 100 people are known
to have contracted this disease, what is the probability that fewer than 30 survive?
2. A multiple choice quiz has 200 questions, each with 4 possible answers of which only 1 is
correct. What is the probability that sheer guesswork yields from 25 to 30 correct answers for
the 80 of the 200 problems about which the student has no knowledge?
3. A coin is tossed 400 times. Use the normal curve approximation to find the probability of
obtaining (a) between 185 and 210 heads inclusive; (b) exactly 205 heads; (c) fewer than 176 or
4. A process yields 10% defective items. If 100 items are randomly selected from the process, what
is the probability that the number of defectives (a) exceeds 13? (b) is less than 8?
5. A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs (a) at least 25
times? (b) between 33 and 41 times inclusive? (c) exactly 30 times?
6. A common practice of airline companies is to sell more tickets for a particular flight than there
are seats on the plane, because customers who buy tickets do not always show up for the flight.
Suppose that the percentage of no-shows at flight time is 2%. For a particular flight with 197
seats, a total of 200 tickets were sold. What is the probability that the airline overbooked this
flight?
7. Researchers at George Washington University and the National Institute of Health claim that
approximately 75% of people believe “tranquilizers work very well to make a person more calm
ENGDATA LECTURE9
Page |6

and relaxed”. Of the next 80 people interviewed, what is the probability that (a) at least 50 are of
this opinion? (b) at most 56 are of this opinion?
8. If 20% of the residents in a U.S. city prefer a white telephone over any other color available,
what is the probability that among the next 100 telephones installed in that city (a) between 170
and 185 inclusive will be white? (b) at least 210 but not more than 225 will be white?
The text passage is a lecture on probability distributions, specifically focusing on the continuous uniform distribution and the normal distribution. It includes multiple examples and questions related to these distributions, but does not provide answers to these questions within the text.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Sie sind auf Seite 1von 6

ENGDATA LECTURE9

Page |1

CONTINUOUS UNIFORM DISTRIBUTION

The continuous uniform distribution is one of the simplest continuous distributions in statistics.
It is characterized by a density function that is flat and the probability is uniform in a closed interval [A,
B]. It is often called the rectangular distribution because the density function forms a rectangle with base
B-A and constant height 1/(B-A).

The density function of the continuous uniform random variable X on the interval [A, B] is

{
1
f ( x ; A , B )= B− A , A ≤ x ≤ B ,
0 , elsewhere .

The mean and variance of the uniform distribution are

2
A+ B (B− A)
μ= ∧σ 2= .
2 12

Examples:
1. Suppose that a large conference room at a certain company can be reserved for no more than 4
hours. Both long and short conferences occur quite often. In fact, it can be assumed that the
length X of a conference has a uniform distribution on the interval [0, 4]. (a) What is the
probability density function? (b)What is the probability that any given conference lasts at least 3
hours?
2. The daily amount of coffee in liters, dispensed by a machine located in an airport lobby is a
random variable X having a continuous uniform distribution with A=7 and B=10. Find the
probability that on a given day the amount of coffee dispensed by this machine will be (a) at
most 8.8 liters; (b) more than 7.4 liters but less than 9.5 liters; (c) at least 8.5 liters.
3. A bus arrives every 10 minutes at a bus stop. It is assumed that the waiting time for a particular
individual is a random variable with a continuous uniform distribution. (a) What is the
probability that the individual waits more than 7 minutes? (b) What is the probability that the
individual waits between 2 and 7 minutes?
4. Suppose X follows a continuous uniform distribution from 1 to 5. Determine the conditional
probability P( X >2.5∨X ≤ 4) .

NORMAL DISTRIBUTION

The normal distribution is the most important probability distribution in statistics.Its graph (bell-
shaped curve) is called the normal curve.The graph of a normal distribution is a bell-shaped curve that
extends indefinitely in both directions.

Ex: physical measurements in meteorological experiments, rainfall studies and measurements of

manufactured parts

 Pierre Laplace (1749-1827)

 Abraham De Moivre (1667-1745)
ENGDATA LECTURE9
Page |2

 Karl Friedrich Gauss (1777-1855)

The density of the normal random variable X, with mean μ and variance σ 2 , is
−1 2
( x−μ )
1 2σ
2
, −∞< x < ∞ ,
n ( x ; μ , σ )= e
√2 π σ
2
The mean and variance of n(x; μ,σ) are μ and σ , respectively. The standard deviation is
σ .

The normal curve (sometimes called standard curve) Normal curves with μ 1< μ 2 and σ1=σ2

Normal curves with μ 1=μ 2 and σ1 < σ2 Normal curves with μ 1< μ 2 and σ1 < σ2

Properties of Normal Curve

 A probability distribution is said to be normal if the mean, median and mode coincides at a
single point.
 The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at
x=μ .
 The curve is symmetric about a vertical axis through the mean μ .
 The curve has its points of inflection at x=μ ± σ ; it is concave downward if
μ−σ < X < μ+ σ and is concave upward otherwise.
ENGDATA LECTURE9
Page |3

 The right and left tails of the normal curve are asymptotic with respect to the horizontal axis.
 The total area under the curve is equal to 1 or 100%.

The Empirical Rule

The 68-95-99.7 Rule of the Normal Curve
 About 68% of the area under the curve falls within 1 standard deviation of the mean.
(−1< z <1)
 About 95% of the area under the curve falls within 2 standard deviations of the mean.
(−2< z<2)
 About 99.7% of the area under the curve falls within 3 standard deviations of the mean.
(−3< z< 3)

AREAS UNDER THE NORMAL CURVE

The curve of any continuous probability distribution or density function is constructed so that
the area under the curve bounded by two ordinates x=x 1 and x=x 2 equals the probability that
the random variable X assumes a value between x=x 1 and x=x 2 .

The distribution of a normal random variable with mean 0 and variance 1 is called a standard
normal distribution.

The z-Score
Fortunately, we are able to transform all the observations of any normal random variable x
to a new set of observations of a normal random variable z . The formula is:
x−μ
z=
σ
where: x = score (observation)
μ = mean
σ = standard deviation
z = z-score

Examples:
1. Given a standard normal distribution, find the area under the curve that lies (a) to the right of
z=1.84 and (b) between z=-1.97 and z=0.86.
2. Given a standard normal distribution, find the value of k such that (a) P(Z>k) = 0.3015 and (b)
P(k<Z<-0.18) = 0.4197.
3. Given a random variable X having a normal distribution with μ=50∧σ =10 , find the
probability that X assumes a value between 45 and 62.
4. Given that X has a normal distribution with μ=300∧σ =50 , find the probability that X
assumes a value greater than 362.
5. Given a normal distribution with μ=40∧σ=¿ 6, find the value of x that has (a) 45% of the
area to the left and (b) 14% of the area to the right.
6. Given a standard normal distribution, find the area under the curve that lies (a) to the right of
z=1.25 ; (b) to the left of z=−0.4 ; (c) to the left of z=0.8 ; (d) between z=0.4
and z=1.3 ; (e) between z=−0.3 and z=0.9 ; and (f) outside z=−1.5 to
z=1.5 .
ENGDATA LECTURE9
Page |4

7. Find the value of z if the area under a standard normal curve (a) to the right of z is 0.3622; (b) to
the left of z is 0.1131; (c) between 0 and z, with z>0, is 0.4838; (d) between –z and z, with z>0, is
0.9500.
8. Given a normal distribution with μ=30∧σ =6 , find (a) the normal curve area to the right of
x=17; (b)the normal curve area to the left of x=22; (c) the normal curve area between x=32 and
x=41; (d) the value of x that has 80% of the normal curve are to the left; € the two values of x
that contain the middle 75% of the normal curve area.

APPLICATIONS OF THE NORMAL DISTRIBUTION

Examples:
1. A certain type of storage battery lasts, on average, 3.0 years with a standard deviation of 0.5
year. Assuming that the battery lives are normally distributed, find the probability that a given
battery will last less than 2.3 years.
2. An electrical firm manufactures light bulbs that have a life, before burn-out, that is normally
distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find the
probability that a bulb burns between 778 and 834 hours.
3. In an industrial process the diameter of a ball bearing is an important component part. The
buyer sets specifications on the diameter to be 3.0 ± 0.01 cm. The implication is that no part
falling outside these specifications will be accepted. It is known that in the process the diameter
of a ball bearing has a normal distribution with mean μ = 3.0 and standard deviation σ = 0.005.
On the average, how many manufactured ball bearings will be scrapped?
4. Gauges are used to reject all components for which a certain dimension is not within
specification 1. 50 ± d . It is known that this measurement is normally distributed with mean
1.50 and standard deviation 0.2. Determine the value d such that the specifications “cover” 95%
of the measurements.
5. The average grade for an exam is 74, and the standard deviation is 7. If 12% of the class is given
As, and the grades are curved to follow a normal distribution, what is the lowest possible A and
the highest possible B? Also, find the sixth decile.
6. A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard
deviation of 2 ohms. Assuming the resistance follows a normal distribution and can be measured
to any degree of accuracy, what percentage of resistors will have a resistance exceeding 43
ohms? Find the percentage of resistances exceeding 43 ohms if resistance is measured to the
nearest ohm.
7. The IQs of 600 applicants of a certain college are approximately normally distributed with a
mean of 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how
many of these students will be rejected on this basis regardless of their other qualifications?
8. A radar unit is used to measure speeds of cars on a motorway. The speeds are normally
distributed with a mean of 90 km/hr. and a standard deviation of 10 km/hr. What is the
probability that a car picked at random is travelling at more than 100 km/hr.?
9. For a certain type of computers, the length of time between charges of the battery is normally
distributed with a mean of 50 hours and a standard deviation of 15 hours. John owns one of
these computers and wants to know the probability that the length of time will be between 50
and 70 hours.
10. A large group of students took a test in Physics and the final grades have a mean of 70 and a
standard deviation of 10. If we can approximate the distribution of these grades by a normal
distribution, what percent of the students (a) scored higher than 80? (b) should pass the test
ENGDATA LECTURE9
Page |5

11. The SAT is a test for readiness of students for college. The average SAT score (on a 1600 point
scale) is 1025 points and the standard deviation is 200 points. How well must Jessica do on the
SAT in order to place in the top 10% of all students?
12. An intelligence quotient, or IQ, is a score derived from one of several standardized tests
designed to assess intelligence. The mean score is normalized as 100 and the standard deviation
is roughly 15. An IQ of 150 is required for entrance into a gifted program, what percentage of
students is considered eligible?
13. At the New Age Information Corporation, the ages of all new employees hired during the last 5
years are normally distributed. Within this curve, 95.4% of the ages, centered about the mean,
are between 24.6 and 37.4 years. Find the mean age and the standard deviation of the data.
14. The heights of 1000 students are normally distributed with a mean of 174.5 centimeters and a
standard deviation of 6.9 centimeters. Assuming that the heights are recorded to the nearest
half-centimeter, how many of these students you would expect to have heights (a) less than
160.0 centimeters? (b) between 171.5 and 182.0 centimeters inclusive? (c) equal to 175.0
centimeters? (d) greater than or equal to 188.0 centimeters?
15. The weights of a large number of miniature poodles are approximately normally distributed with
a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements are recorded
to the nearest tenth of a kilogram, find the fraction of these poodles with eights (a) over 9.5
kilograms; (b) of at most 8.6 kilograms; (c) between 7.3 and 9.1 kilograms inclusive.

NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

2
If X is a binomial random variable with mean μ=np and variance σ =npq , then the
limiting form of the distribution of
X−np
Z= As n → ∞ , is the standard normal distribution n( z ; 0,1) .
√ npq
Examples:
1. The probability that a patient recovers from a rare blood disease is 0.4. If 100 people are known
to have contracted this disease, what is the probability that fewer than 30 survive?
2. A multiple choice quiz has 200 questions, each with 4 possible answers of which only 1 is
correct. What is the probability that sheer guesswork yields from 25 to 30 correct answers for
the 80 of the 200 problems about which the student has no knowledge?
3. A coin is tossed 400 times. Use the normal curve approximation to find the probability of
obtaining (a) between 185 and 210 heads inclusive; (b) exactly 205 heads; (c) fewer than 176 or
4. A process yields 10% defective items. If 100 items are randomly selected from the process, what
is the probability that the number of defectives (a) exceeds 13? (b) is less than 8?
5. A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs (a) at least 25
times? (b) between 33 and 41 times inclusive? (c) exactly 30 times?
6. A common practice of airline companies is to sell more tickets for a particular flight than there
are seats on the plane, because customers who buy tickets do not always show up for the flight.
Suppose that the percentage of no-shows at flight time is 2%. For a particular flight with 197
seats, a total of 200 tickets were sold. What is the probability that the airline overbooked this
flight?
7. Researchers at George Washington University and the National Institute of Health claim that
approximately 75% of people believe “tranquilizers work very well to make a person more calm
ENGDATA LECTURE9
Page |6

and relaxed”. Of the next 80 people interviewed, what is the probability that (a) at least 50 are of
this opinion? (b) at most 56 are of this opinion?
8. If 20% of the residents in a U.S. city prefer a white telephone over any other color available,
what is the probability that among the next 100 telephones installed in that city (a) between 170
and 185 inclusive will be white? (b) at least 210 but not more than 225 will be white?
user
The text passage is a lecture on probability distributions, specifically focusing on the continuous uniform distribution and the normal distribution. It includes multiple examples and questions related to these distributions, but does not provide answers to these questions within the text.

TYPE: MULTIPLE_QUESTIONS