CH 4 Overview

1
CH 4 Overview

• Ch 4 combines the methods of descriptive
  statistics presented in ch 2 and those of
  inferential statistics or probability presented
  in ch 3

2
Random Variables (Ch4-2)

• Random variable? Its is an x-value determined by
  chance for each outcome of a procedure.
• Probability distribution? Its a table or graph
  or formula that gives the probability for each
  value of x. Table 4-1 P183
• Discrete Variable? It has a finite or countable
  value.
• Continuous variable? It has infinitely many
  values. Example p184

3
Requirements for Probability Distribution

• There is no probability distribution without the
  following 2 requirements
• SP(x) 1, meaning that the sum of all
  probabilities equal 1
• 0ltP(x)lt1, meaning that no probability should be
  less than 0, neither grater tha1.
• Example p185

4
Mean, Variance, and Standard Deviation

• Lets go back to CVDOT learned in ch 2 and use
  the following formulas to find the mean, variance
  and standard deviation for a probability
  distribution
• µ SxP(x) mean
• s2 S(x - µ)2 P(x) variance
• s2 Sx2 P(x) - µ2 variance
• ? s2 Sx2 P(x) - µ2 standard deviation
• Example pp188189

5
Unusual Results

• As we did in section 2.5 for the range, most
  values should fall within 2 standard deviations
• Max. Unusual Value µ 2s
• Min. Unusual Value µ - 2s
• As we also said, this rule is not perfect but it
  can help in finding unusual values. Anything
  outside the range could be checked for unusual.
  Example PP188-189

6
Unusual Results and Probability

• When the unusual value is high, its probability
  is low.
• Example p190

7
Expected Value

• Expected value is very important in Decision
  Theory. It is the average value of the outcomes.
• E x p(x) This can help in finding the
  expected value for an infinite many trials.
• Example p191

8
Binomial Probability Distributions

• Binomial probability distribution is very
  important because it deals with circumstances in
  which the outcomes belong to two relevant
  categories such as good/bad, yes/no, boys/girls,
  etc
• P(x) n! (n x)!x! px q(n-x)
• n N. of trials
• x N. of successes in n trials
• p Probability of success in one of the n trials
  
• q Probability of failure in one of the n trials
  
• p(x) Probability of getting exactly x successes
  
• Please read the requirements p 196 and Example
  Pp199200

9
Mean, Variance, Standard Deviation for Binomial
Distribution

• Lets go back to what we just did in section 4-2
  to find these values again using now Binomial
  Distribution formulas
• µ Sxp(x) gt µ np
• s2 Sx2 p(x) - µ2 gt s2 npq
• s ? Sx2 P(x) - µ2 gt s ?(npq)
• Example Pp208209

10
Poisson Distribution (Sect. 4-5)

• The poisson distribution is not needed for the
  rest of this course, but you still need to
  understand it because its a very important
  mathematical model.
• P(x) µxeµx! where e 2.7182.
• Please Read Pp213-215
• Suggested HW PP192-194 1-12, 15-18
• PP203-206 1-24, 33, 34

11
Test 2 Review Answers