Ôn tập Xác Suất Thống Kê

Compute the mean number of aircraft accidents per month.b.. Compute the probability of no accidents during a month.

Chapter 4: Probability

a) A and B are mutually exclusive, If A happen then B can’t happen

P (A or B) = P (A ⋃ B) = P(A) + P(B)

(Probability of events union)

b) A and B have intersection

Multiplication law: P (A or B) = P (A ⋃ B) = P(A) +P(B) - P (A ⋂ B)

P (A ⋂ B) = P (A and B) = Events intersection

c) Conditional Probability

P (A | B) =

*Independent events

P (A | B) = P(A)

P (B | A) = P(B)

d) ML for independent events

P (A ⋂ B) = P(A) P(B)

E.g:

P(A)= 6/10 P(B)=4/10

P(D)=1/2 P(C)= ½

P (C and B) =1/10

P (A or D)= P(A) + P(D) – P(A and D) = 6/10+1/2-2/10

P( B and D)= 3/10

Chapter 5: Discrete Varieable

Function:

1) Expected Value= np

2) Variance= E [] -E=np(1-p)

3) Binomial Function

Keywords: Probality of 2 choices YES AND NO

X have n trials and probability p

 X ~ B(n,p)

Select K times for X

4) Poisson Probability Function

Keyword: Average

Function : f(x)=

F(x)= the probability of x occurrences in an interval

µ: expected value of the mean of occurrences in an interval

e= 2,71828

Examples: An average of 15 aircraft accidents occur each year (The World Almanac and Book of Facts, 2004)

a Compute the mean number of aircraft accidents per month.

b Compute the probability of no accidents during a month

Solution:

Chapter 6 Continuous Probability Distributions

Type 1: Uniform Distribution

Uniform Probability Function: f(x) = (a ≤ x ≤ b); otherwise f(x)=0

Type of math

Keywords: uniform distribution from a to b

Ask: Probability of a range (c;d)=> Compute f(x) => compute the length of the range by

d-c=> P(c ≤ x ≤ d)= f(x).range(d;c)

Type 2: Normal Distribution Function

Z= with ;

Keywords: Normal distribution, the probability distribution is normal, are normally

distributed

ASK:

1) The probability of a range (a;b), Find z and P

=>x1=a =>z1

=>x2=b => z2

We have z1< z < z2

 P(z2) – P(z1)= P (z1< z <z2)

2) From z and P Find x

Top of a%=> P= a% =>z=? =>x=?

Type 3: Normal Approximation of Binomial Probabilities

Condition for this function: np ≥ 5; n(1-p) ≥ 5

Standard deviation =

Mean= np

ASK: Given n= a and p= b

Solution: Check the given condition => compute the mean and standard deviation=>

Compute Z => From z we will have P

Situation 1: probability of c

o c-0.5 < x < c+0.5

o compute the range of z-score

o P(c+0.5) – P (c-0.5)

Situation 2: probability of fewer than c => c+0.5

Situation 3: probability of more than c => c-0.5

Situation 4: the range from d to e

 D-0.5 < X < E+0.5 Chapter 8 Estimation and Confidence Interval

a) σ known

ME= with n is number of sample

b) σ unknown

Using t-table instead of z-table

E= with s is sample standard deviation and t-score, and df= n-1 (degree of freedom)

E=

c) Sample size

n=

d) With proportion π

E=

Chapter 9: Hypothesis Test

Step 1: Determine the case

Case 1: : µ≥ ; : µ < (Lower tail)

Case 2: : µ ≤ ; : µ > (Upper tail)

Case 3: : µ = ; : µ ≠ (Two tail)

Step 2: Compute the Statistical test

1) Z-score=

2) p-value based on the tail of case

3) Rejection Rule

Chapter 10: 2 population hypothesis