Review Slides

# Review Slides - Chapter 3 Probability-Counting Techniques...

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Chapter 3: Probability-Counting Techniques General Counting Rules The Addition Rule: Suppose event 1 has p outcomes and event 2 has q outcomes. Then event 1 or 2 (but not both) have p+q outcomes. The Multiplication Rule : Suppose event 1 has p outcomes and the unrelated event 2 has q outcomes. Then together there are p×q possible outcomes. Permutation Objects are drawn sequentially or ordered from left to right in a row. (Order matters; objects are drawn without replacement). ( 1)( 2). ..(2)(1) ! 0! 1 n n n n with - - = = ( ) (0) ! ( 1)( 2). ..( 1) with n 1 ( )! r n n n n n r n n r - - - + = = = - 1 2 ! ! !... ! r n n n n ( ) ! ( 1). ..( 1)( )( 1). ..(2)(1) !( )! !( )( 1). ..(2)(1) ! r n n n n n r n r n r n r r n r r n r n r r   - - + - - - = = =   - - - -   Combinations

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Union of two events: ( ) ( ) ( ) - ( ) Union of two events mutually exclusive: ( ) ( ) ( ) Intersection: ( ) ( ) Complement: ( ) 1 ( ) ( ) Conditional Probability: ( | ) ( ) Indepe P A B P A P B P AB P A B P A P B P A B P AB P A P A P AB P A B P B ∪ = + ∪ = + = = - = I ndence: P(AB)=P(A)P(B) or P(A)P(B) P(A|B)= ( ) P(B) For mutually exclusive events: P(AB)=0 P A = Chapter 4:Probability Rules and Conditional Probability
The multiplication rule: P(AB)=P(B)P(A|B) The Partition Rule: P(B)=P(BA 1 )+P(BA 2 )+…+P(BA k ) = P(A 1 )P(B|A 1 )+…P(A k )P(B|A k ) Bayes Theorem: Chapter 4:Probability Rules and Conditional Probability ( ) ( | ) ( ) ( | ) ( ) ( | ) ( ) ( ) ( | ) ( ) ( | ) ( ) P AB P B A P A P B A P A P A B P B P B P B A P A P B A P A = = = +

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Random Variable is a function that assigns a real number to each point in a S There are discrete and continuous random variables Probability function: Describe the probabilities associated with a random variable f(x)=P(X=x), for all x A Properties: 1. f(x)≥0 for all values of x (i.e. for x A) 2. ∑ all x f(x)=1 Cumulative distribution function: (CDF): Is the function F(x)= P(X≤x), for x A . Properties of a c.d.f. F(x): 1. F(x) is a non-decreasing function of x 2. 0≤F(x)≤1 for all x A 1 ) ( lim and 0 ) ( lim . 3 = = -∞ x F x F x x Chapter 5: Discrete Random Variables and Probability Models
: Suppose X takes values a,a+1,a+2, ,b with all values being equally likely. Then X has a discrete uniform distribution, on {a,a+1,a+2, ,b}. Probability Function: 5 Chapter 5: Discrete Random Variables and Probability Models HYPERGEOMETRIC DISTRIBUTION N objects can be classified into two types (S and F). There are total of r (S) and N-r (F) in the population. Pick sample of n objects at random without replacement . X

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## This note was uploaded on 10/21/2010 for the course STAT 230 taught by Professor Various during the Fall '06 term at Waterloo.

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Review Slides - Chapter 3 Probability-Counting Techniques...

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