Bài giảng thống kê ứng dụng trong quản lý xây dựng Lê Hoài Long

Bài giảng thống kê ứng dụng trong quản lý xây dựng Lê Hoài Long 11Population and sample 12Variables 13Measures of data 14Pattern of data 15Scales of measurement 16Between variables 21Data collection 22Probability 23Variable 24Sampling distribution 31One sample estimation 32One sample hypothesis testing 33Two sample 34Multisample estimation and testing 35Nonparametric techniques

Part 2 – Section 3

VARIABLE AND DISTRIBUTION

 When the numerical value of a variable is

determined by a chance event, that variable

is called a random variable

 Random variables can be discrete or

continuous

 A probability distribution is a table or an

equation that links each possible value that a random variable can assume with its

probability of occurrence

 Two types:

 Discrete Probability Distributions

 Continuous Probability Distributions

Discrete Probability Distributions

 The probability distribution of a discrete

random variable can always be represented

by a table

 Given a probability distribution, you can find cumulative probabilities

Continuous Probability Distributions

 The probability distribution of a continuous random

variable is represented by an equation, called the

probability density function (pdf).

 All probability density functions satisfy the following

conditions:

 The random variable Y is a function of X; that is, y = f(x).

 The value of y is greater than or equal to zero for all

values of x.

 The total area under the curve of the function is equal to one.

Continuous Probability Distributions

 The probability that a continuous

random variable falls in the interval

between a and b is equal to the area

under the pdf curve between a and b.

 There are an infinite number of

values between any two data points

As a result, the probability that a

continuous random variable will

assume a particular value is always

Mean of a Discrete Random Variable

 The mean of the discrete random variable X is

also called the expected value of X, denoted by

E(X):

 where xi is the value of the random variable for outcome i, μx is the mean of random variable X, and P(xi) is the probability that the random

variable will be outcome i.

Median of a Discrete Random Variable

 The median of a discrete random variable is the "middle" value

 It is the value of X for which P(X < x) is

greater than or equal to 0.5 and P(X > x) is

greater than or equal to 0.5

 The number of hits made by each player is

described by the following probability distribution Number of hits, x 0 1 2 3 4

Probability, P(x) 0.10 0.20 0.30 0.25 0.15

 What is the mean of the probability distribution?

 What is the median?

Variability of a Discrete Random Variable

 The equation for computing the variance of a discrete random variable is:

Sums and Differences of Random Variables:

 If X and Y are random variables, then

where E(X) is the expected value (mean) of X, E(Y) is the expected value of Y, E(X + Y) is the expected value of X plus Y, and E(X - Y) is the expected value of X minus Y.

) (

) (

) (

) (

) (

) (

Y E X

E Y

X E

Y E

X E

Y X

Sums and Differences of Random Variables:

 The variance of (X + Y) and the variance of (X Y) are described by the following equations

-where Var(X + Y) is the variance of the sum of X and Y, Var(X - Y) is the variance of the

difference between X and Y, Var(X) is the

) (

) (

) (

)

Independence of Random Variables

 If two random variables, X and Y, are

independent, they satisfy the following

conditions (either one)

Y and X

of values all

for )

( )

( )

(

Y and X

of values all

for )

( )

(

y P x

P y

x

P

x P y

 Considering only these types:

BINOMIAL DISTRIBUTION

Binomial Experiment

 The experiment consists of n

repeated trials

 Each trial can result in just

two possible outcomes

(success and failure)

 The probability of success,

denoted by p, is the same

on every trial

 Notation

 x: The number of successes that result from the binomial

experiment

 n: The number of trials in the binomial experiment

 p: The probability of success on an individual trial

 q: The probability of failure on an individual trial.

 f(x): Binomial probability function

 nCx: The number of combinations of n things, taken x at a

Binomial Distribution

 A binomial random variable is the number

of successes x in n repeated trials of a

binomial experiment

 The probability distribution of a binomial

random variable is called a binomial

distribution (also known as a Bernoulli

distribution).

The binomial distribution has the following

properties:

 The mean of the distribution (μx) is equal to np

 The variance (σ 2

x) is np(1 - p).

 The standard deviation is σx

 It is generally agreed that if the ratio of

sample size to population size is no more

than 0.05, the trials without replacement are essentially independent

Binomial Probability

 The binomial probability refers to the

probability that a binomial experiment results

in exactly x successes Suppose a binomial experiment consists of n trials and results in

x successes If the probability of success on

an individual trial is P, then the binomial

x

n C p p x

f ( )  ( 1  ) 

Cumulative Binomial Probability

 A cumulative binomial probability refers to

the probability that the binomial random

variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper

x n x

x n

a x

p p

C x

f a

Example: Flooding of a road.

Suppose a road is flooded with probability p = 0.1 during a year and not more than one flood

occurs during a year

 What is the probability that it will be flooded at least once during a 5-year period?

 What is the probability that it will be flooded

twice during a 5-year period?

 What is the probability distribution?

Applied: Lot-acceptance sampling:

- A quality control procedure

- A standard is set and if the products meet

this standard, then they are accepted

- A lot is a large number of the same items so impossible to test all

- Therefore, a much smaller random sample is taken from each lot to test

Applied: Lot-acceptance sampling

- For a given sampling plan with parameters n

(sample size) and r (acceptance number, e.g

defective items in sample), the probability that a lot will be accepted is Pr (with p is the expected proportion in the lot)

x n x

r

p

p x

n x

n

r F r

x P P

) 1

( )!

(

!

!

) ( )

(

- The sample plan: sample size n=10 items

and defective item not larger than 1

Applied: Lot-acceptance sampling

n = 15

So, with various n

(sample size) selected, please specify the resulted

effects.

Applied: Lot-acceptance sampling

Consumer’s risk and producer’s risk

 Consumer’s risk is the probability of accepting a lot that has a higher p than the consumer can tolerate

 Producer’s risk is the probability of a lot being rejected that is actually in conformity with the

consumer' The

P



 1 risk

s producer' The

NEGATIVE BINOMIAL

Negative Binomial Experiment

 The experiment consists of x repeated trials

 Each trial can result in just two possible

outcomes

 The probability of success, denoted by p, is the

same on every trial

 The trials are independent

 The experiment continues until r successes are observed, where r is specified in advance

 f*(x): Negative binomial probability

 nCr: The number of combinations of n things,

taken r at a time.

Negative Binomial Distribution

 A negative binomial random variable is the

number x of repeated trials to produce r

successes in a negative binomial

experiment

 The probability distribution of a negative

binomial random variable is called a

negative binomial distribution (Pascal

Negative Binomial Probability

 The negative binomial probability refers to

the probability that a negative binomial

experiment results in r - 1 successes after

trial x - 1 and r successes after trial x

r x r

r

x

f * ( )  1 1 ( 1  ) 

The Mean of the Negative Binomial

Distribution

 If we define the mean of the negative binomial

distribution as the average number of trials required

to produce r successes, then the mean is equal to:

where μ is the mean number of trials, r is the

number of successes, and p is the probability of a success on any given trial.

p

r





Example: Delivery of equipments.

A company has bid to supply equipments for a system in a region, having quoted a low price for the job However, the supervising engineer has estimated from previous

experience that 10% of equipments by this company are

defective in someway If 5 equipments are required,

determine the minimum number of equipments to be

ordered to be 95% sure that a sufficient number of

nondefective equipments are delivered It is assumed that the delivery of an equipment is an independent trial and any fault that may occur in one equipment is not related to

GEOMETRIC DISTRIBUTION

Geometric Distribution

 The geometric distribution is a special

case of the negative binomial distribution

 It deals with the number of trials required for

a single success Thus, the geometric

distribution is negative binomial distribution

where the number of successes (r) is equal

to 1

Geometric Probability Formula.

 Suppose a negative binomial experiment

consists of x trials and results in one

success If the probability of success on an

individual trial is p, then the geometric

probability is:

1

) 1

( )

( x  p  p xf

MULTINOMIAL DISTRIBUTION

MULTINOMIAL EXPERIMENT

 Consisting of n identical trials

 For each trial, there are k

possible mutually exclusive

events (A1, …, Ak)

 The probability of Ai is pi and

pi remains constant over trials

 Sum of pi equals unity

 The trials are independent

MULTINOMIAL PROBABILITY FUNCTION

 The multinomial probability function gives the probability in the n trials of multinomial

experiment:

k

x k x

k

x x

n x

,

1 1

MULTINOMIAL DISTRIBUTION

Example: Bids for contracts.

A city engineer invites separate bids for widening four roads Three contractors submit their quotations The first contractor is usually successful in getting 60% of similar work in the area, where as the other two have equal chances of 15%.

What is the probability that the first contractor will be given at least three of the jobs on the basis of past performances?

HYPERGEOMETRIC DISTRIBUTION

HYPERGEOMETRIC EXPERIMENT

 Resembling the binomial experiment except that the hypergeometric involves sampling

from a finite population without replacement

 A random sample of n objects is taken one at a time from a finite population of NT objects by

sampling without replacement

 Of the NT objects, NS are of one type, called

‘successes’, and NF are of another type

 The random variable X is used to count the

number of successes

(

!)!

(

!

!

)(

N

x n

N x

n

N x

N x

N

C

C

C x

f

T

F

F S

S

n N

x n N

x N

T

F s

HYPERGEOMETRIC PROBABILITY

MEAN, VARIANCE AND SD

E ( ) 

) 1 (

T F

S

N N

n N

HYPERGEOMETRIC DISTRIBUTION

Example: Personnel organization

 A manager must select a committee of three from his staff of six men (M) and four women (W) He writes their names on separate

pieces of paper, puts in a bowl, then blindly picks a sequence of three papers

 Find the probability that he picks two W?

POISSON DISTRIBUTION

1 For a given continuous unit of time or space,

there is a known empirically determined

positive constant, denoted by , that is the

average rate of occurrence of successes in the given unit

2 For any size of subunit of the given unit, the

number of successes occurring in the subunit

is independent of the number of successes in any other nonoverlapping subunit

3. If the specified unit is divided into very

small subunits denoted by t, the probability

of exactly one success occurring in an t is

very small and it is the same for all ts in the

unit no matter when (where) they appear

4. The probability of more than one success

occurring in any very small subunit t is

essentially zero

 The poisson probability function utilizes the constant t to determine the probability of

occurrence of successes (X) in some

multiple t of the defining unit for a Poisson

experiment

!

)

( )

( )

(

x

e

t x

X P

x f

POISSON DISTRIBUTION

 The mean:

 The variance:

 The standard deviation:

t X

 The cumulative Poisson probability values

calculated with this equation:

t x

x

e

t a

F

!

)

( )

(





 A manufacturer of cable, knowing that defects appear

‘randomly’ in the cable as it is produced, wants to use

Poisson techniques to determine the probabilities for

different numbers of defects in a fixed length (unit) of

cable He decides to use 4 meters as the fixed unit and after counting defects in many 4-meter lengths, he finds that the average of there counts is 4 defects per 4 meters.

 Find the average rate of occurrence of defects in 1 meter

 Find the probability of two defects occurring in 1 meter

 Floorboards supplied by a contractor have some imperfections A builder decides that two

imperfections per 40m2 is acceptable

 Is there at least a 95% chance of meeting such

requirements, if from previous experience with the same material, an average of one imperfection per

65 m2 has been found?

 We deal with some types:

 If a continuous random variable

X can assume any value in the

interval [a,b] and only these

values, and if its probability

density function f(X) is constant

over that interval and equal to

zero elsewhere, then X is said to

be uniformly distributed.

elsewhere

0 )

(

for

,

1 )

b x

a a

b X

f

 An industrial psychologist has determined

that it takes a worker between 9 and 15

minutes to complete a task on an assembly line If the continuous random variable – time

to complete the task – is uniformly distributed over the interval [9,15]

 Finding the probability that a worker will

complete a task shorter than 13 minutes

 A continuous random

variable X is said to be

exponentially distributed if,

for any >0, its probability

density function is given by

the exponential probability

density function:

0for x

0f(x)

0for x

)

 The cumulative distribution function:

0for x

0

0for x

1

)(

)(

X P

x

Example: Floods affecting construction.

 An engineer constructing a bridge across a river is concerned of the possible occurrence of a flood

exceeding 100 m3/s which can seriously affect his work If a flow of such magnitude is exceeded once

in 5 years on average, on the basis of recorded

data, what is the chance that the work which is

scheduled to last 14 months can proceed without interruption or detrimental effects?

 The beta distribution models a random variable that takes values in the interval given by 0–1.

 The distribution plays a special role in decision methods.

1 1

) 1

( )

, (

0 )

(

0 ,

) 1

( )

, (

1 )

, ,

(

dx x

x B

x

f

x

x B

and 1

x 0

for

( X

E

) 1 (

Example: Maintenance of major roads.

There are 10 major roads in province A and a similar

number and length of roads in province B The

proportion of roads that require substantial

maintenance works during an annual period can be

approximated by beta(4, 3) and beta(1, 4) distributions, respectively, in the two provinces.

(1) Which province should spend more on annual

maintenance?

(2) What is the probability that not more than two roads will require substantial maintenance work in province B during an annual period?

 The value of the random variable Y is:

where X is a normal random variable, μ is the

mean, σ is the standard deviation





x

e Y