Engineering Mathematics II

ENGINEERING MATHEMATICS II

010.141

STATISTICAL

NUMERICAL SOLUTIONS

MODULES 7

STATISTICAL NUMERICAL SOLUTIONS

Ø 24.1 Data

Ø 24.2 Experiments, outcomes, events Ø 24.3 Probability

Ø 24.5 Random variables, and Probability distribution Ø 24.6 Mean and variance

Ø 24.7 Binomial, Poisson, and Hypergeometric Ø 24.8 Normal

Ø 24.9 Distribution of several random variables

DATA REPRESENTATION

Ø Average; spread Ø Histograms

Ø Center; median; quartile Ø Box plot

Ø Outlier

Ø Mean; standard deviation; variance.

ILLUSTRATIONS

Ø Consider these 14 umbers:

78, 81, 83, 84, 86, 87, 87, 89, 89, 89, 89, 90, 91, 99

Ø These are arranged in order ; half-way is between 87 and 89; the median is half-way, so the median would be either 87 or 89.

Ø The numbers could be grouped as follows:

§ A: 77 +/- 2.5 A: one number

§ B: 82 +/- 2.5 B: 3

§ C: 87 +/- 2.5 C: 7

§ D: 92 +/- 2.5 D: 2

§ B: 97 +/- 2.5 E: 1

EXAMPLE OF HISTOGRAM

Number by interval

A B C D E

6 7

4 5

2 3

0 1

SOME DEFINITIONS

Ø Absolute frequency: number of times a value shows up

Ø Relative frequency: number of times a value shows up, divided by total of numbers

Ø Cumulative absolute frequency: running total Ø Range: from min value to max value

Example: 78 to 91 is a range of 21

Ø Median: data value that falls in the middle, when lined up in order;

in current example, between 87 and 89; use 88 as median

Ø Outlier: value that seems to be different from the other values in the set.

MORE DEFINITIONS

Ø Mean: average of the data; sum of the values, divided by the number of values.

Indicated by (x - bar) Sometimes called the arithmetic mean

In the running example, there are 14 numbers; the sum of the values is 611, so the mean is 611/14, or 87.3.

Note that the mean is slightly below the median, 88

VARIANCE AND STANDARD DEVIATION

Ø The variance indicates spread in the data

Ø The definition of variance of a data set, labeled s² is:

Ø In the current example, s² = 25.14

Ø The standard deviation, s, is: s = 5.014

( )

s = 1

n 1 x x

2

j j = 1

-

å

- ²

n

SECTION 24.2

EXPERIMENTS, OUTCOMES, EVENTS

Ø Trial: single performance of an experiment Result is an outcome or sample point n trials yields a sample of size n

Sample space is the set of all possible outcomes

Ø Set union: The union of two sets is a set whose elements are in either set, or both sets

Ø Set intersection: The intersection of two sets is a set whose elements are in both sets (intersection might be empty).

Ø Set complement: another set whose elements are in the sample space, but not the first set.

ROLL OF THE DICE

Ø Two dice are rolled; there are six numbers, 1 to 6, on each; the sum of the numbers could be as low as 2 or as high as 12. The dice are assumed to be fair. The next chart depicts the 36 outcomes. Note that the number 7 shows up as the sum six times, and the number 11 shows up twice. If either 7 or 11 is a winner on the roll, then the likelihood of winning on this first roll is 8/36, or 22%.

Suppose that rolling 2, 3, or 12 is considered a loser. There are four such cases in the table. The likelihood of losing on the first roll is thus 4/36, or 11%.

Note that the likelihood of winning is twice losing, on the first roll at least.

OUTCOMES OF ROLLS OF DICE

Ø Roll of two dice and sample space of outcomes of sum of two dice.

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

SECTION 24.3 PROBABILITY

Ø Definition 1: probability

§ Pr(A) = (number of points in A) / (number of points in S)

§ Pr(S) = 1

§ 0 ≤ Pr ≤ 1

Ø Relative frequency of A:

§ f_rel(A) = f(A)/n

§ relative frequency lies between 0 and 1

§ relative frequency of S is 1

§ if two sets A and B are mutually exclusive (intersection is empty) then relative frequency of the union of A and B is the sum of their relative frequencies

Pr(A È B) = Pr(A) + Pr(B)

PROBABILITY

Ø Definition 2. Given sample space S; event A in S (A is a subset of S), there are some axioms of probability:

§ 0 ≤ Pr ≤ 1

§ Pr(S) = 1

§ Pr(A È B) = Pr(A) + Pr(B),

for the case where A and B are non-overlapping (that is, their intersection is null)

ADDITION RULE

Ø Given events A and B in a sample space

Pr(A È B) = Pr(A) + Pr(B) - Pr(A Ç B) Note that if A and B are exclusive, the last term is 0 The probability of the null set is zero

Conditional probability of B, given A:

Pr(B|A) = Pr(B Ç A) / Pr(A) where Pr(A) not = 0

( ) ( )

( ) ( )

Pr A|B = Pr A B

Pr BI Pr B 0

; ¹

MULTIPLICATION RULE

Suppose in a sample space S there are two events A and B and their probabilities are greater than zero.

Then:

Pr(A Ç B) = Pr(A) · Pr(B|A) = Pr(B) · Pr(A|B) Conditional Probability is labeled A|B

Ø Independence: If A and B are independent, then:

Pr(A Ç B) = Pr(A) · Pr(B)

SECTION 24.5

Ø Define:

F(x) = Pr(X ≤ x) F(x) is called a distribution function

Pr(a < X ≤ b) = F(b) - F(a) Ø A probability function could be defined:

Density function

In this case it is called discrete; the distribution function is thus:

RANDOM VARIABLES

PROBABILITY DISTRIBUTIONS

( )

f x p_j x

= = x otherwise

j

0 ìí î

( ) ( )

F x =

å

f x = _j

å

p_j

A random variable X and its distribution function F(x) is continuous provided it can be represented by:

the function f is referred as the density or density function;

As before,

It is necessary that the integral over all possible values be unity.

CONTINUOUS

( ) ( )

F x = f d

x

n n

-¥

ò

( ) ( )

Pr a < x b = f d

a b

£

ò

^{n n}

( ) ( )

f x = F x¢

SECTION 24.6

MEAN AND VARIANCE OF A DISTRIBUTION

Consider a uniform distribution (see next page), where f(x) = 1/(b - a), on a < x < b; otherwise = 0.

( )

( )

( ) ( )

( ) ( )

( )

m

m

s m

s m

s s

= x

= x

= x

= x

> 0 Note: could be zero

i i

2

i i

2

f x

f x dx

f x

f x dx

j

j

å ò

å ò

-¥

¥

-¥

¥

-

-

2

2 2

2

UNIFORM DISTRIBUTION

Height = 1/(b - a); Area = 1

The red curve illustrates a more informed density function. The uniform

40 90

70 ?

A B

MEAN AND VARIANCE OF UNIFORM DISTRIBUTION

( ) ( )

( )

( )

( )

b 2 2 2

a

b 2 3

2 a

3 3

3

x x b a a + b

= = = =

b a 2 b a 2 b a 2

a + b 1 1 a + b

= x = x

2 b a 3 b a 2

1 a + b a + b

= b a

3 b a 2 2

1 b a

= 3 b a 2

b

a

b

a

dx

dx m

s

-

- - -

æ ö æ ö

- × × -

ç ÷ ç ÷

- -

è ø è ø

ì ü

ïæ ö æ ö ï

×íç - ÷ - ç - ÷ ý

- ïîè ø è ø ïþ

æ - ö

× ç ÷ -

- è ø

ò ò

( )

( )

( ) ( )

( )

3

3 2 2 3 3 2 2 3

3

2 2

a b 2

= 1 3ab + 3a b a a + 3a b 3ab + b 24 b a

1 2 b a

24 b a b a = 12

b

s

ì - ü

ï æ ö ï

í ç ÷ ý

è ø

ï ï

î þ

× - - - -

-

= × -

- -

SECTION 24.7

BINOMIAL, POISSON,

HYPERGEOMETRIC DISTRIBUTIONS

Ø Binomial: how often does event A occur in n independent trials?

Let p = probability of success per trial and q failure; p + q = 1 (also, q is the same as "not p")

Let X = number of times that A occurs in n trials for X = x, the probability is:

( )

n

x p q = n!

n x p q

x n x x n x

æ èç ö

ø÷

-

- -

x! !

RATIONALE FOR BINOMIAL

For x successes in n trials (n - x) failures

one arrangement is p · p · p · · · p · q · q · · · q with probability p^x q^n-x

for all possible permutations use Theorem 1(b), page 1065:

Probability of x successes in n trials

( )

n!

n x p q = n

x p q

x n x x n x

x! - !

æ èç ö

ø÷

- -

MEAN AND VARIANCE OF BINOMIAL

Ø Mean

μ = np Ø Variance

σ² = npq If

( )

2

p = q = 1 flip a coin 2

= n 2 = n

4

m

s

DISCRETE POISSON DISTRIBUTION

( )

f x = x e x = 0, 1, 2, = 0! = 1

x 2

m ^m s m

! ^- L,

Example 2, page 1081

p = 0.01 n = 100 μ = np = 1 0 ≤ x ≤ 100

Pr .

Pr .

Pr . ob

ob

ob

x = 0 = 1

0! e = e = 36%

x = 1 = 1

1! e = e = 36%

x = 2 = 1

2! e = 1

2 e = 18%

0

1

2

- -

- -

- -

1 1

1 1

1 1

Ø Suppose that the probability that an item produced by a machine will be defective is 0.1. What is the probability that a sample often items, selected at random from the output of the machine, will have no more than one defective?

Answer from binomial:

Answer from Poisson:

EXAMPLE - BINOMIAL VS. POISSON

( ) ( ) ( ) ( )

10

0 01 0 9 10

1 01 0 9

0 10 1 9

æ èç ö

ø÷ æ

èç ö ø÷

. . + . . = 0.7361

( )( )

m = np = 10 0.1 = 1 e + e = 0.7358^{-1 -1}

SECTION 24.8

Ø Definition of normal distribution

Ø Integral form

NORMAL DISTRIBUTION

( )

f x = 1

2 exp x

s p

m - æ s-

èç ö

ø÷ é

ë ê ê

ù û ú ú 1

2

2

( )

F x = 1 d

2 exp

x

s p

n m

s n

- æ - èç ö

ø÷ é

ë ê ê

ù û ú

-¥

ò

2¹ ú

2

PROPERETIES OF THE NORMAL DISTRIBUTION

Ø Mean μ Ø Variance σ²

For standard normal

( ) ( )

f z

z d

z

= 1 2 e

= 1 2

e

x

z

p

f

p

m

m

m -

-

-¥

ò

-

2

2

2

2

EXAMPLE

( )

m s s

m s

m s

= 50 = 9 = 3 Find c such that Pr x < c = 5%

x = c 50

8, page A90 for 5%

x = 1.645

c 50

= 1.645 c = 45.065

2

- -

- -

- -

3

3

See Table _95%

45 47 56

44 x = 50 53

: = 50

EXAMPLE

( )

= 0.01 = 0.001

Probability that R is between 0.009 and 0.011 at lower end

x

= 0.009 0.01

0.001 = 0.001

0.001 = 1.000 x

= 0.011 0.01

0.001 = 0.001

0.001 = 1.000 89

1 = 0.8413 0.1587 = 0.6826 = 68%

For 1000 wires 683

088, figure 489(a)

m s

m s

m s

f

- - -

-

- -

-

® See page A

See page 1

GRAPH OF THE NORMAL DENSITY FUNCTION

( )

( ) ( )

f

p

f

p x = 1

2 e Standard Normal

x = y = 1

2 e dy

x -

-¥

-

-¥

ò ò

1 2

1 2

2

2

x

x y

dy F

GRAPH OF THE NORMAL DENSITY FUNCTION

N(x)

0.242 0.319

0.058 0.014

-4 -3

-4 3 4

-1.96 -0.67 2.58

-2.58

50% of area 68.3% of area

95% of area

0.67 1.96

1 2 -2 -1

GRAPH OF THE NORMAL DISTRIBUTION FUNCTION

N(x)

0.2

-3 -2 -1 0 ^{0.25 0.67} 1 2 3 4

-4

0.4 0.6 0.7 0.75 0.8 0.9 0.95

1.0

0.52 0.84 1.28 1.96

0.5