Ôn tập Xác Suất Thống Kê

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= 2) Variance= np E [] -E=np(1-p) 3) Binomial Function Keywords: Probality of 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 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 P= a% =>z=? =>x=? Type 3: Normal Approximation of Binomial Probabilities Condition for this function: np ≥ 5; n(1-p) ≥ 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 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: population hypothesis