CHAPTER 1.

PROBABILITY

I. Solution Steps

+ Step 1: Define the events from the problem.

+ Step 2: Define the probabilities and conditional probabilities for the events defined in Step 1.

+ Step 3: Find the system of events which is both mutually exclusive and collectively exhaustive (compute the complement if needed).

+ Step 4: Apply the formula.

II. Probability rules

( )
1. Complement rule: P A = 1 - P ( A) .

2. Addition rule: P ( A È B ) = P ( A) + P ( B ) - P ( A Ç B ) .

In particular, if A, B are mutually exclusive, then P ( A È B ) = P ( A) + P ( B ) .

P ( A Ç B)
3. Conditional probability: P ( A | B ) = .
P ( B)

4. Multiplication rule: P ( A Ç B ) = P ( A) P ( B | A) = P ( B ) P ( A | B )

Events A, B are said to be statistically independent if and only if P ( A Ç B ) = P ( A) P ( B ) .

5. Total probability:

Given the system of events A1 , A2 ,..., An that are both mutually exclusive and collectively exhaustive, then

n
P( B) = å P( Ai ) P( B Ai )
i =1

= P ( A1 ) P ( B | A1 ) + P ( A2 ) P ( B | A2 ) + ... + P ( An ) P ( B | An )

( ) (
or P ( B ) = P ( A) P ( B | A) + P A P B | A )
 CHAPTER 2. DISCRETE RANDOM VARIABLES

I. Distribution table

Given the distribution table:

X x1 x2 …. xn

P p1 p2 …. pn

Then

ì0 £ pi £ 1
ï
1. í n .
ïå pi = 1
î i =1

2. P(a < X < b) = å

a < xi <b
pi .

n
3. µ = E ( X ) = å pi xi .
i =1

æ n ö
4. s 2 = Var ( X ) = ç å xi2 pi ÷ - µ 2 .
è i =1 ø

In particular, if Y = aX + b then µY = aµ X + b; s Y2 = a 2s X2 .
II. Binomial Distribution

Suppose that

• a random experiment can result in two possible outcomes, “success” and “failure,”
• and that p is the probability of a success in a single trial.

Let X be the the number of resulting successes in n independent trials.

The probability distribution of X is called binomial distribution.

P( X = k ) = Cnk p k (1 - p)n-k

E ( X ) = np

Var ( X ) = np(1 - p)

III. Poisson Distribution

Let X be the number of occurrences in a given continuous interval (such as time, surface area, or length). Then the probability distribution
of X is called the Poisson distribution.

e- l l k
P( X = k ) = ,
k!

E ( X ) =V (X ) = l
 CHAPTER 3. CONTINUOUS RANDOM VARIABLES

I. Density Function

Given the density function f ( x ) of a continuous random variable X . Then

i) f ( x ) ³ 0, "x Î R

+¥
ii) ò f ( x ) dx = 1
-¥

b
iii) P ( a £ X < b ) = P ( a < X £ b ) = P ( a £ X £ b ) = P ( a < X < b ) = ò f ( x ) dx
a

x
iv) F ( x ) = ò f ( t ) dt
-¥

+¥
v) E ( X ) = ò x f ( x)dx
-¥

+¥
vi) V ( X ) = ò ( x - E ( X ))
2
f ( x)dx = 𝑥 ! . 𝑓 (𝑥 ) 𝑑𝑥 − 𝜇!
-¥
II. Normal Distribution

If X follows the normal distribution with the mean µ and variance s 2 , then

æb-µ ö æa-µ ö
P ( a < X < b) = F ç ÷-Fç ÷
è s ø è s ø

The values of F ( x ) can be found in Appendix Table 1 with the notices

i) F ( - x ) = 1 - F ( x ) .

ii) x > 4 : F ( x ) = 1

iii) x < -4 : F ( x ) = 0

( ) ( )
Moreover, if X  N µ X , s X2 and Y  N µY , s Y2 , then aX + bY also follows the normal distribution

aX + bY  N ( a µ X + bµY ; a 2s X2 + b 2s Y2 )
 CHAPTER 4. SAMPLING DISTRIBUTION

Let the random variables X1 , X 2 ,..., X n denote a random sample from a population.

I. Sample mean

X 1 + X 2 + ... + X n
- Sample mean: X = .
n

- If the parent population distribution is normal (or n ³ 25 ) then

æ s2 ö X -µ
X  N ç µ; ÷ or Z =  N ( 0; 1) .
è n ø s / n

II. Sample proportion

X
- Sample proportion: p =
n

where n is the sample size and X is the number of objects having the characteristic of interest.

- When n is large ( nP (1 - P ) > 5 ), we have

p  N æ P; P (1 - P ) ö or Z =
p - P
ç ÷  N ( 0; 1)
è n ø P (1 - P )
n
III. Sample variance

(X ) +(X ) ( )
2 2 2
1- X 2 - X + ... + X n - X
- Sample variance: S 2 = .
n -1

- If the parent population distribution is normal then

( n - 1) S 2  c n2-1
s 2
 CHAPTER 5. CONFIDENCE INTERVAL ESTIMATION

I. Formulas

Confidence Interval Estimation with the sample n and the confidence level (1 - a )
Population mean µ Population mean µ Population proportion P Population variance s 2

Given
Estimation Given Given Given
for ( ) (
+ X  N µ , s (or n is + X  N µ , s 2 (or n
2
) is + n is large (
+ X  N µ ,s 2 )
large) large)
+ s is known
2
+ s 2 is unknown
Confidence x ± za /2
s
n
x ± tn-1, a /2
s
n p ± z (
p 1 - p
) æ ( n - 1) s 2 ( n - 1) s 2
çç 2 ; 2
ö
÷÷
interval a /2
n è n-1, a /2 c n-1,1-a /2
c ø
Margin of ME = za /2
s
n
ME = tn-1,a /2
s
n ME = za /2
(
p 1 - p
) æ
ME = ( n - 1) s 2 ç 2
1
- 2
1 ö
ç c n-1,1-a /2 c n-1, a /2 ÷÷
error è ø
n

Width
w = 2 za /2
s
n
w = 2 tn-1, a /2
s
n w = 2 za /2
(
p 1 - p
) æ
w = 2 ( n - 1) s 2 ç 2
1
- 2
1 ö
ç c n-1,1-a /2 c n-1, a /2 ÷÷
n è ø
s s
( ) ( n - 1) s 2
Upper p 1 - p
UCL = x + za /2 UCL = x + tn-1,a /2 UCL = 2
confidence n n UCL = p + za /2 c n-1,1-a /2
limit n
Lower
confidence
LCL = x - za /2
s
n
LCL = x - tn-1, a /2
s
n LCL = p - za /2
(
p 1 - p
) LCL =
( n - 1) s 2
c n2-1,a /2
limit n
II. Critical values

1. Critical values for the standard normal distribution:

za is a value such that P(Z > za ) = a

Some critical values: z0,05 = 1,645; z0,025 = 1,96; z0,01 = 2,33; z0,005 = 2,58

Given the critical value za , we can determine the significant level a (or confidence level 1 - a ) by the following formula

a = 1 - F ( za )

where F ( za ) can be found in Appendix Table 1.

2. Critical values for the Student’s t distribution

tn-1,a is a value such that P(Tn-1 > tn-1,a ) = a

The value tn-1, a can be found in Appendix Table 8.

3. Critical values for the Chi-squared distribution

cn2-1,a is a value such that P( cn2-1 > cn2-1,a ) = a

The value c n2-1,a can be found in Appendix Table 7a and 7b.

 CHAPTER 6. HYPOTHESIS TESTS

I. Solution steps

+ Step 1: State the null and alternative hypotheses.

+ Step 2: Summarize all information.

+ Step 3: Decision rule: reject H 0 if …… (Formula)

+ Step 4: Random sample => conclusion

II. Decistion rule

ì H 0 : µ = µ0 ì H 0 : µ = µ0 ì H 0 : µ = µ0
For population mean í í í
î H1 : µ ¹ µ0 î H1 : µ > µ0 î H1 : µ < µ0
s
Reject H 0 if x < µ0 - za /2 s s
X  N ( µ ,s 2
) (or n is large) n Reject H 0 if x > µ0 + za Reject H 0 if x < µ0 - za
s n n
s 2 is known or x > µ0 + za /2
n
s
Reject H 0 if x < µ0 - tn-1, a /2
X  N ( µ ,s ) (or n is large)
2 s s
n Reject H 0 if x > µ0 + tn-1, a Reject H 0 if x < µ0 - tn-1, a
s n n
s 2 is unknown or x > µ0 + tn-1, a /2
n
For population ì H 0 : P = P0 ì H 0 : P = P0 ì H 0 : P = P0
í í í
proportion î H1 : P ¹ P0 î H1 : P > P0 î H1 : P < P0
P0 (1 - P0 ) Reject H 0 if Reject H 0 if
Reject H 0 if p < P0 - za /2
nP0 (1 - P0 ) ³ 5
n P0 (1 - P0 ) P0 (1 - P0 )
p > P0 + za p < P0 - za
P0 (1 - P0 ) n n
or p > P0 + za /2
n

For population ïH0 : s = s 0

ì 2 2
ìï H 0 : s 2 = s 02 ìï H 0 : s 2 = s 02
í í í
variance î H1 : s ¹ s 0
ï
2 2
ïî H1 : s > s 0
2 2
ïî H1 : s < s 0
2 2

( n - 1) s 2 > c n2-1, a /2
Reject H 0 if
s 2 ( n - 1) s 2 > c n2-1, a
( n - 1) s 2 < c 2
X  N ( µ ,s 2 ) 0 Reject H 0 if Reject H 0 if n -1,1- a
s 02 s02

or
( n - 1) s 2
<c 2
n -1,1-a /2
s 02